The Supercontinuum Laser Source: Fundamentals with Updated References

The Supercontinuum Laser Source Second Edition

Robert R. Alfano Editor

The Supercontinuum Laser Source Fundamentals with Updated References Second Edition

With 259 Illustrations

Robert R. Alfano Distinguished Professor of Science and Engineering Department of Physics City College of the City University of New York New York, NY 10031 USA [email protected]

Cover illustration: Shows the supercontinuum generation of intensity versus wavelength for 1 mm of carbon tetrachloride liquid excited by a 120-fs, 625-nm laser pulse. Photo by Robert R. Alfano, A. Katz, and P.P. Ho. Library of Congress Cataloging-in-Publication Data The supercontinuum laser source: fundamentals with updated references/[edited by] Robert R. Alfano. p. cm. Includes bibliographical references and index. ISBN 0-387-24504-9 (acid-free paper) 1. Laser pulses, Ultrashort. 2. Nonlinear optics. I. Alfano, Robert R., 1941– QC689.5.L37S87 2005 621.36¢6—dc22 2005042765 ISBN-10: 0-387-24504-9 ISBN-13: 978-0387-24504-1

e-ISBN 0-387-25097-2

Printed on acid-free paper. © 2006, 1989 Springer Science+Business Media, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, Inc., 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed in the United States of America. 9 8 7 6 5 4 3 2 1 springeronline.com

(BS/MVY)

To my father, Alfonso L. Alfano and my father-in-law, Samuel J. Resnick whose advice I deeply miss.

Preface to the Second Edition

The “supercontinuum” (SC) has become one of the hottest topics to study in optical and photonic sciences since the first book on the supercontinuum was published, entitled The Supercontinuum Laser Source, by Springer in 1989. That book, now becoming Part I in this second edition, reviewed the progress achieved on the experimental and theoretical understanding of the ultrafast nonlinear and linear processes responsible for the supercontinuum generation and related applications occurring over 20 years since its discovery by Robert R. Alfano and Stanley Shapiro in 1969. There is a great need for a sequel part covering the recent worldwide surge of research activity on the supercontinuum phenomena and the numerous technological applications that have occurred over the past 15 years. This void will partly be covered in this new rejuvenated second edition, called Part II, by an overview of the recent advances with an updated compendium of references on the various breakthroughs to understand the supercontinuum and its new diverse applications. The supercontinuum is the generation of intense ultrafast broadband “white-light” pulses spanning the ultraviolet to the near infrared that arises from the nonlinear interaction and propagation of ultrafast pulses focused into a transparent material. The supercontinuum can be generated in different states of matter—condensed media (liquids and solids) and gases. The supercontinuum is one of the most dramatic and elegant effects in optical physics. The conversion of one color to white-light is a startling result. This is multicolored light with many of the same desirable properties as conventional laser light: intense, collimated, and coherent. The supercontinuum has a beam divergence as good as that of the input pump laser pulse. Moreover, the coherence length of the supercontinuum is comparable with that of an incoherent white-light source from a light bulb. The interference pattern measured for the supercontinuum from a pair of filaments in water shows a constant phase relationship between the supercontinuum produced by each filament. There is a constant phase relationship between the pump laser pulse and its supercontinuum. The white-light supercontinuum is an ideal tunable ultrafast white-light laser source. Supercontinuum has overtaken the study of

viii

Preface to the Second Edition

other nonlinear optical effects such as second harmonic generation (SHG) and two-photon absorption for usefulness in a number of diverse applications. The supercontinuum field is still active after 36 years, and is today finding new and novel uses. Various processes are involved in the supercontinuum generation. Whenever an intense ultrashort laser pulse propagates through a medium, it changes the refractive index from the distortion of the atomic and molecular configuration, which in turn changes the phase, amplitude, and frequency of the incident pulse. The phase change and amplitude change can cause a frequency sweep of the carrier wave within the pulse envelope and can alter the envelope and spatial distribution (self-focusing). There are various mechanisms responsible for the index of refraction change in material with intensity. The frequency broadening mechanisms are electronic cloud distortion, reorientational, librations, vibrational, and molecular redistribution, to name the major ones. The operation of these mechanisms depends on its relaxation time relevant to the laser pulse duration. The relaxation times associated with electronic distribution is of the order of Bohr orbit time ~150 as; reorientation time is ~1 ps; rocking and libration response about the field is ~1 ps; vibrational dephasing is ~0.1 ps; and molecular motion is ~1 ps. Most of these mechanisms are involved in the supercontinuum generation with 100 fs to ps laser pulses. Soon after the supercontinuum discovery in 1969, it initially found applications in time-resolved pump-supercontinuum probe absorption and excitation spectroscopy to study the fundamental picosecond (10-12 s) and femtosecond (10-15 s) processes that occur in biology, chemistry, and solidstate physics. Briefly, in biology, the primary events in photosynthesis and vision were explored; in chemistry, a better understanding of the basic chemical dynamical steps in reactions and nonradiative processes in photoexcited chemicals was achieved; and in solid-state physics, the underlying kinetics of how elementary excitations behave and relax, such as optical phonons, polaritons, excitons, and carriers (electrons and holes) dynamics among the intervalleys and intravalley of semiconductors, were unraveled. With the advent of microstructure fibers, there has been a rebirth of the supercontinuum field in the type of applications in which the supercontinuum can play a decisive role. These applications include frequency clocks, phase stabilization and control, timing, optical coherence tomography (OCT), ultrashort pulse compression, optical communication, broad spectrum LIDAR, atmospheric science, lighting control, attosecond (10-18 s) pulse generation, and coherence control. Over the past several years, supercontinuum generation in microstructure photonic crystal fibers by ultrashort pulse propagation has become a subject of great interest worldwide. The main reasons are the low pulse energies required to generate the supercontinuum; its coherences and high brightness makes the continuum an ideal white-light source for diverse applications; and the effects of zero dispersion and anomalous dispersion regions has resulted

Preface to the Second Edition

ix

in higher-order solutions generation, pulse compression, and an ultrabroadband continuum exceeding 1000 nm, extending from the ultraviolet to the infrared spectral regions. In microstructural fibers, when pump wavelength lies in an anomalous dispersion region, it is the solitons that initiate the formation of the continuum. In a normal dispersion region, self-phase modulation is the process that initiates the continuum generation. The combination of four-wave mixing and Raman processes extends the spectral width of the continuum. In that regard, the pulse duration of an ultrafast laser determines the operational mechanisms—for 10 fs to 1000 fs laser pulses, self-phase modulation and soliton generation dominates; and for pulses >30 ps, stimulated Raman and fourwave mixing play a major role in extending the spectra. Of course, the pump wavelength location, relative to the zero dispersion wavelength and the anomalous dispersion region, plays a role in the active mechanism and coherence region of the supercontinuum. The supercontinuum spectra can span more than a two-optical octave bandwidth spread from 380 nm to 1600 nm using 200 fs pulses with energy in the tens of nanojoules. The span over an octave (i.e., 450 nm to 900 nm) is important in controlling the phase of the carrier wave inside the pulse envelope of a mode-locked pulse train. Using the f and 2f waves in the supercontinuum, the carrier-envelope offset (CEO) phase can be detected using heterodyne beating between the high-frequency end of the supercontinuum with the doubled low end frequency of the supercontinuum in an interferometer. These phase-controlling effects are important for maintaining the accuracy of frequency combs for clocking and timing in metrology, high-intensity atomic studies, and attosecond pulse generation. The increasing worldwide demand for large-capacity optical communication systems needs to incorporate both the wavelength and time. The ultrabroad bandwidth and ultrashort pulses of the supercontinuum may be the enabling technology to produce a cost-effective superdense wavelength division multiplexing (>1000 l) and time multiplexing for the future Terabits/s to Pentabits/s communication systems and networks. The supercontinuum is an effective way to obtain numerous wavelength channels because it easily generates more than 1000 optical longitudinal modes while maintaining their coherency. The propagation of ultrahigh power femtosecond pulses ~100 GW (10 mJ at 100 fs) in “air” creates the supercontinuum from the collapse of the beam by self-focusing into self-guided small-size filaments. These filament tracks in air are more or less stable over long distances of a few kilometers due to the balance between self-focusing by the nonlinear index of refraction (n2) and the defocusing by the ionized plasma formation via multiphoton ionization. The supercontinuum in air can be used to monitor the amount of trace gases and biological agents in aerosols in the backscattering detection geometry for LIDAR applications. Furthermore, remote air ionization in the atmosphere by the intense femtosecond pulses in the filaments plasma (uses the supercontinuum as the onset marker) has the potential to trigger, control, and

x

Preface to the Second Edition

guide lightning from one point to another and possibly even induce condensation by seeding clouds to make rain. This approach may be able to secure and protect airports and power stations from lightning and may be used to collect and store energy from lightning. Moreover, creating an ionized filament track in a desirable region may be used to confuse and redirect the pathway of incoming missiles for defense. This new second edition will consist of two parts. The major portion (Part I) of the new book will be the reprinting of Chapters 1 to 10 from the first edition. These chapters lay down the understanding and foundation of the birth of the supercontinuum field. They go over the salient experimental and theoretical concepts in the research works produced up to 1989. The second part of this new second edition includes a new chapter (Chapter 11) highlighting the supercontinuum coherence and 10 additional chapters (Chapters 12 to 21) listing updated references of papers on the recent advances made in our understanding and applications of supercontinuum. These papers will be referenced and arranged within a topical group where a brief overview of the key features of these papers within a topic will be presented. The following are the selected topics to be highlighted in the new Chapters 12 to 21 of updated references: • Supercontinuum generation in materials (solids, liquids, gases, air). • Supercontinuum generation in microstructure fibers. • Supercontinuum in wavelength division multiplex telecommunication. • Femtosecond pump—supercontinuum probe for applications in semiconductors, biology, and chemistry. • Supercontinuum in optical coherence tomography. • Supercontinuum in femtosecond carrier-envelope phase stabilization. • Supercontinuum in ultrafast pulse compression. • Supercontinuum in time and frequency metrology. • Supercontinuum in atmospheric science. • Coherence of the supercontinuum. Special thanks to Ms. Lauren Gohara and Dr. Kestutis Sutkus for their assistance in the production of the second edition. New York, New York

Robert R. Alfano

Preface to the First Edition

This book deals with both ultrafast laser and nonlinear optics technologies. Over the past two decades, we have seen dramatic advances in the generation of ultrafast laser pulses and their applications to the study of phenomena in a variety of fields. It is now commonplace to produce picosecond (10-12 s) pulses. New developments have extended this technology into the femtosecond (10-15 s) time region. Soon pulses consisting of just a single cycle will be produced (i.e., 2 fs at 600 nm). These ultrafast pulses permit novel investigations to study phenomena in many disciplines. Sophisticated techniques based on these laser pulses have given rise to instruments with extremely high temporal resolution. Ultrafast laser technology offers the possibility of studying and discovering key processes unresolved in the past. A new era of timeresolved spectroscopy has emerged, with pulses so fast that one can now study the nonequilibrium states of matter, test quantum and light models, and explore new frontiers in science and technology. Ultrashort light pulses are a potential signal source in future high-bit-rate optical fiber communication systems. The shorter the pulses, the more can be packed into a given time interval and the higher is the data transmission rate for the tremendous bandwidth capacity of optical fiber transmission. Nonlinear optics is an important field of science and engineering because it can generate, transmit, and control the spectrum of laser pulses in solids, liquids, gases, and fibers. One of the most important ultrafast nonlinear optical processes is the supercontinuum generation—the production of intense ultrafast broadband “white-light” pulses—that is the subject of this book. The first study on the mechanism and generation of ultrafast supercontinuum dates back over 19 years to 1969, when Alfano and Shapiro observed the first “white” picosecond pulse continuum in liquids and solids. Spectra extended over ~6000 cm-1 in the visible and infrared wavelength region. They attributed the large spectral broadening of ultrafast pulses to self-phase modulation (SPM) arising from an electronic mechanism and laid down the formulation of the supercontinuum generation model. Over the years, the improvement of mode-locked lasers led to the production of wider super-

xii

Preface to the First Edition

continua in the visible, ultraviolet, and infrared wavelength regions using various materials. The supercontinuum arises from the propagation of intense picosecond or shorter laser pulses through condensed or gaseous media. Various processes are responsible for continuum generation. These are called self-, induced-, and cross-phase modulations and four-photon parametric generation. Whenever an intense laser pulse propagates through a medium, it changes the refractive index, which in turn changes the phase, amplitude, and frequency of the pulse. However, when two laser pulses of different wavelengths propagate simultaneously in a condensed medium, coupled interactions (cross-phase modulation and gain) occur through the nonlinear susceptibility coefficients. These coupled interactions of two different wavelengths can introduce phase modulation, amplitude modulation, and spectral broadening in each pulse due to the other pulse using cross-effects. An alternative coherent light source to the free electron laser, the supercontinuum laser source, can be wavelength selected and coded simultaneously over wide spectral ranges (up to 10,000 cm-1) in the ultraviolet, visible, and infrared regions at high repetition rates, gigawatt output peak powers, and femtosecond pulse durations. Ultrafast supercontinuum pulses have been used for time-resolved absorption spectroscopy and material characterization. Supercontinuum generation is a key step for the pulse compression technique, which is used to produce the shortest optical pulses. Future applications include signal processing, three-dimensional imaging, ranging, atmospheric remote sensing, and medical diagnosis. Thus far, a great deal of information on supercontinuum technology has been obtained and has enhanced our understanding of how intense optical pulses propagate in materials. These developments are most often found in original research contributions and in review articles scattered in journals. Textbooks do not cover these subjects in great detail. There is a need for a book that covers the various aspects of ultrafast supercontinuum phenomena and technology. This book reviews present and past progress on the experimental and theoretical understanding of ultrafast nonlinear processes responsible for supercontinuum generation and related effects such as pulse compression and ultrashort pulse generation on a picosecond and femtosecond time scale. The content of the chapters in the book is a mixture of both theoretical and experimental material. Overviews of the important breakthroughs and developments in the understanding of supercontinuum during the past 20 years are presented. The book is organized into 10 chapters. Summarizing the highlights of the 10 chapters of the book: In Chapter 1, Shen and Yang focus on the theoretical models and mechanisms behind supercontinuum generation arising mainly from self-phase modulation.

Preface to the First Edition

xiii

In Chapter 2, Wang, Ho, and Alfano review the experiments leading to the supercontinuum generation in condensed matter over the past 20 years. In Chapter 3, Agrawal discusses the effects of dispersion on ultrafast light pulse propagation and supercontinuum generation in fibers. In Chapter 4, Baldeck, Ho, and Alfano cover the latest experimental observations and applications of the cross-interactions in the frequency, time, and space domains of strong pulses on weak pulses. In Chapter 5, Manassah reviews the theoretical models giving rise to many phenomena from self-phase and induced modulations. In Chapter 6, Suydam highlights the effect of self-steepening of pulse profile on continuum generation. In Chapter 7, Corkum and Rolland review the work on supercontinuum and self-focusing in gaseous media. In Chapter 8, Glownia, Misewich, and Sorokin utilize the supercontinuum produced in gases for ultrafast spectroscopy in chemistry. In Chapter 9, Dorsinville, Ho, Manassah, and Alfano cover the present and speculate on the possible future applications of the supercontinuum in various fields. In Chapter 10, Johnson and Shank discuss pulse compression from the picosecond to femtosecond time domain using the continuum and optical dispersive effects of gratings, prisms, and materials. The reader will find that these chapters review the basic principles, contain surveys of research results, and present the current thinking of experts in the supercontinuum field. The volume should be a useful source book and give young and seasoned scientists, engineers, and graduate students an opportunity to find the most necessary and relevant material on supercontinuum technology in one location. I hope these efforts will stimulate future research on understanding the physics behind supercontinuum technology and exploring new applications. I wish to thank all the expert contributors for their cooperation in this endeavor. Most thought it would not be completed. Special thanks goes to Mrs. Megan Gibbs for her administrative and secretarial assistance. I gratefully acknowledge T. Hiruma for his continued support. I pay particular tribute to my friend Stan Shapiro, who missed seeing the outgrowth of our first work in this field 20 years ago. New York, New York

Robert R. Alfano

Contents

Preface to the Second Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preface to the First Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii xi xix

Part I

Fundamentals

Chapter 1

Theory of Self-Phase Modulation and Spectral Broadening Y.R. Shen and Guo-Zhen Yang . . . . . . . . . . . . . . . .

1

Supercontinuum Generation in Condensed Matter Q.Z. Wang, P.P. Ho, and R.R. Alfano . . . . . . . . . . .

33

Ultrashort Pulse Propagation in Nonlinear Dispersive Fibers Govind P. Agrawal . . . . . . . . . . . . . . . . . . . . . . . . .

91

Cross-Phase Modulation: A New Technique for Controlling the Spectral, Temporal, and Spatial Properties of Ultrashort Pulses P.L. Baldeck, P.P. Ho, and R.R. Alfano . . . . . . . . . .

117

Simple Models of Self-Phase and Induced-Phase Modulation Jamal T. Manassah . . . . . . . . . . . . . . . . . . . . . . . . . .

184

Self-Steepening of Optical Pulses B.R. Suydam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

295

Self-Focusing and Continuum Generation in Gases Paul B. Corkum and Claude Rolland . . . . . . . . . . .

318

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 6

Chapter 7

xvi

Contents

Chapter 8

Chapter 9

Chapter 10

Utilization of UV and IR Supercontinua in Gas-Phase Subpicosecond Kinetic Spectroscopy J.H. Glownia, J. Misewich, and P.P. Sorokin . . . . . .

337

Applications of Supercontinuum: Present and Future R. Dorsinville, P.P. Ho, Jamal T. Manassah, and R.R. Alfano . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

377

Pulse Compression in Single-Mode Fibers: Picoseconds to Femtoseconds A.M. Johnson and C.V. Shank . . . . . . . . . . . . . . . . .

399

Part II

New Developments

Chapter 11

Coherence Properties of the Supercontinuum Source I. Zeylikovich and R.R. Alfano . . . . . . . . . . . . . . . .

453

Supercontinuum Generation in Materials (Solids, Liquids, Gases, Air) . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Updated References . . . . . . . . . . . . . . . . . . . . . . . .

473 473 473

Supercontinuum Generation in Microstructure Fibers . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Updated References . . . . . . . . . . . . . . . . . . . . . . . .

481 481 482

Supercontinuum in Wavelength Division Multiplex Telecommunication . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Updated References . . . . . . . . . . . . . . . . . . . . . . . .

498 498 498

Femtosecond Pump: Supercontinuum Probe for Applications in Semiconductors, Biology, and Chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Updated References . . . . . . . . . . . . . . . . . . . . . . . .

505 505 506

Supercontinuum in Optical Coherence Tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Updated References . . . . . . . . . . . . . . . . . . . . . . . .

510 510 510

Chapter 12

Chapter 13

Chapter 14

Chapter 15

Chapter 16

Contents

xvii

Supercontinuum in Femtosecond Carrier-Envelope Phase Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Updated References . . . . . . . . . . . . . . . . . . . . . . . .

512 512 514

Chapter 18

Supercontinuum in Ultrafast Pulse Compression . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Updated References . . . . . . . . . . . . . . . . . . . . . . . .

517 517 517

Chapter 19

Supercontinuum in Time and Frequency Metrology . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Updated References . . . . . . . . . . . . . . . . . . . . . . . .

519 519 519

Chapter 20

Supercontinuum in Atmospheric Science . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Updated References . . . . . . . . . . . . . . . . . . . . . . . .

522 522 522

Chapter 21

Coherence of the Supercontinuum . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Updated References . . . . . . . . . . . . . . . . . . . . . . . .

524 524 525

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

529

Chapter 17

Contributors

Govind P. Agrawal, Institute of Optics, University of Rochester, Rochester, New York 14627, USA R.R. Alfano, Department of Physics, City College of the City University of New York, New York 10031, USA P.L. Baldeck, Department of Electrical Engineering, City College of the City University of New York, New York 10031, USA Paul B. Corkum, Division of Physics, National Research Council of Canada, Ottawa, Ontario K1A OR6, Canada R. Dorsinville, Department of Electrical Engineering, City College of the City University of New York, New York 10031, USA J.H. Glownia, IBM Research Division, Thomas J. Watson Research Center, Yorktown Heights, New York 10598, USA P.P. Ho, Department of Electrical Engineering, City College of the City University of New York, New York 10031, USA A.M. Johnson, AT&T Bell Laboratories, Holmdel, New Jersey 07733, USA Jamal T. Manassah, Department of Electrical Engineering, City College of the City University of New York, New York 10031, USA J. Misewich, IBM Research Division, Thomas J. Watson Research Center, Yorktown Heights, New York 10598, USA Claude Rolland, Division of Physics, National Research Council of Canada, Ottawa, Ontario K1A OR6, Canada

xx

Contributors

C.V. Shank, AT&T Bell Laboratories, Holmdel, New Jersey 07733, USA Y.R. Shen, Department of Physics, University of California, Berkeley, California 94720, USA P.P. Sorokin, IBM Research Division, Thomas J. Watson Research Center, Yorktown Heights, New York 10598, USA B.R. Suydam, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA Q.Z. Wang, Department of Physics, City College of the City University of New York, New York 10031, USA Guo-Zhen Yang, Institute of Physics, Academy of Sciences, Beijing, China I. Zeylikovich, Department of Physics, City College of the City University of New York, New York 10031, USA

Part I

Fundamentals

1

Theory of Self-Phase Modulation and Spectral Broadening Y.R. Shen and Guo-Zhen Yang

1. Introduction Self-phase modulation refers to the phenomenon in which a laser beam propagating in a medium interacts with the medium and imposes a phase modulation on itself. It is one of those very fascinating effects discovered in the early days of nonlinear optics (Bloembergen and Lallemand, 1966; Brewer, 1967; Cheung et al., 1968; Lallemand, 1966; Jones and Stoicheff, 1964; Shimizu, 1967; Stoicheff, 1963). The physical origin of the phenomenon lies in the fact that the strong field of a laser beam is capable of inducing an appreciable intensity-dependent refractive index change in the medium. The medium then reacts back and inflicts a phase change on the incoming wave, resulting in self-phase modulation (SPM). Since a laser beam has a finite cross section, and hence a transverse intensity profile, SPM on the beam should have a transverse spatial dependence, equivalent to a distortion of the wave front. Consequently, the beam will appear to have self-diffracted. Such a selfdiffraction action, resulting from SPM in space, is responsible for the wellknown nonlinear optical phenomena of self-focusing and self-defocusing (Marburger, 1975; Shen, 1975). It can give rise to a multiple ring structure in the diffracted beam if the SPM is sufficiently strong (Durbin et al., 1981; Santamato and Shen, 1984). In the case of a pulsed laser input, the temporal variation of the laser intensity leads to an SPM in time. Since the time derivative of the phase of a wave is simply the angular frequency of the wave, SPM also appears as a frequency modulation. Thus, the output beam appears with a self-induced spectral broadening (Cheung et al., 1968; Gustafson et al., 1969; Shimizu, 1967). In this chapter we are concerned mainly with SPM that leads to spectral broadening (Bloembergen and Lallemand, 1966; Brewer, 1967; Cheung et al., 1968; Lallemand, 1966; Jones and Stoicheff, 1964; Shimizu, 1967; Stoicheff, 1963). For large spectral broadening, we need a strong SPM in time (i.e., a large time derivative in the phase change). This obviously favors the use of short laser pulses. Consider, for example, a phase change of 6p occurring in 10-12 s. Such a phase modulation would yield a spectral broadening of

2

Y.R. Shen and G.-Z. Yang

~100 cm-1. In practice, with sufficiently intense femtosecond laser pulses, a spectral broadening of 20,000 cm-1 is readily achievable by SPM in a condensed medium, which is essentially a white continuum (Alfano and Shapiro, 1970). The pulse duration of any frequency component (uncertainty limited) in the continuum is not very different from that of the input pulse (Topp and Rentzepis, 1971). This spectrally superbroadened output from SPM therefore provides a much needed light source in ultrafast spectroscopic studies—tunable femtosecond light pulses (Busch et al., 1973; Alfano and Shapiro, 1971). If the SPM and hence the frequency sweep in time on a laser pulse are known, then it is possible to send the pulse through a properly designed dispersive delay system to compensate the phase modulation and generate a compressed pulse with little phase modulation (Treacy, 1968, 1969). Such a scheme has been employed to produce the shortest light pulses ever known (Fork et al., 1987; Ippen and Shank, 1975; Nakatsuka and Grischkowsky, 1981; Nakatsuka et al., 1981; Nikolaus and Grischkowsky, 1983a, 1983b). Self-phase modulation was first proposed by Shimizu (1967) to explain the observed spectrally broadened output from self-focusing of a Q-switched laser pulse in liquids with large optical Kerr constants (Bloembergen and Lallemand, 1966; Brewer, 1967; Cheung et al., 1968; Jones and Stoicheff, 1964; Lallemand, 1966; Shimizu, 1967; Stoicheff, 1963). In this case, the spectral broadening is generally of the order of a hundred reciprocal centimeters. Alfano and Shapiro (1970) showed that with picosecond laser pulses, it is possible to generate by SPM a spectrally broadened output extending over 10,000 cm-1 in almost any transparent condensed medium. Self-focusing is believed to have played an important role in the SPM process in the latter case. In order to study the pure SPM process, one would like to keep the beam cross section constant over the entire propagation distance in the medium. This can be achieved in an optical fiber since the beam cross section of a guided wave should be constant and the self-focusing effect is often negligible. Stolin and Lin (1978) found that indeed the observed spectral broadening of a laser pulse propagating through a long fiber can be well explained by the simple SPM theory. Utilizing a well-defined SPM from an optical fiber, Grischkowsky and co-workers were then able to design a pulse compression system that could compress a laser pulse to a few hundredths of its original width (Nakatsuka and Grischkowsky, 1981; Nakatsuka et al., 1981; Nikolaus and Grischkowsky, 1983a, 1983b). With femtosecond laser pulses, a strong SPM on the pulses could be generated by simply passing the pulses through a thin film. In this case, the beam cross section is practically unchanged throughout the film, and one could again expect a pure SPM process. Fork et al. (1983) observed the generation of a white continuum by focusing an 80-fs pulse to an intensity of ~1014 W/cm2 on a 500-mm ethylene glycol film. Their results can be understood by SPM along with the self-steepening effect (Manassah et al., 1985, 1986; Yang and Shen, 1984).

1. Theory of Self-Phase Modulation and Spectral Broadening

3

Among other experiments, Corkum et al. (1985) demonstrated that SPM and spectral broadening can also occur in a medium with infrared laser pulses. More recently, Corkum et al. (1986) and Glownia et al. (1986) have independently shown that with femtosecond pulses it is even possible to generate a white continuum in gas media. The phase modulation induced by one laser pulse can also be transferred to another pulse at a different wavelength via the induced refractive index change in a medium. A number of such experiments have been carried out by Alfano and co-workers (1986, 1987). Quantitative experiments on spectral superbroadening are generally difficult. Self-focusing often complicates the observation. Even without self-focusing, quantitative measurements of a spectrum that is generated via a nonlinear effect by a high-power laser pulse and extends from infrared to ultraviolet are not easy. Laser fluctuations could lead to large variations in the output. The simple theory of SPM considering only the lower-order effect is quite straightforward (Gustafson et al., 1969; Shimizu, 1967). Even the more rigorous theory including the higher-order contribution is not difficult to grasp as long as the dispersive effect can be neglected (Manassah et al., 1985, 1986; Yang and Shen, 1984). Dispersion in the material response, however, could be important in SPM, and resonances in the medium would introduce pronounced resonant structure in the broadened spectrum. The SPM theory with dispersion is generally very complex; one often needs to resort to a numerical solution (Fischen and Bischel, 1975; Fisher et al., 1983). It is possible to describe the spectral broadening phenomenon as resulting from a parametric wave mixing process (in the pump depletion limit) (Bloembergen and Lallemand, 1966; Lallemand, 1966; Penzkofer, 1974; Penzkofer et al., 1973, 1975). In fact, in the studies of spectral broadening with femtosecond pulses, four-wave parametric generation of new frequency components in the phasematched directions away from the main beam can be observed together with the spectrally broadened main beam. Unfortunately, a quantitative estimate of spectral broadening due to the parametric process is not easy. In the presence of self-focusing, more complication arises. Intermixing of SPM in space and SPM in time makes even numerical solution very difficult to manage, especially since a complete quantitative description of self-focusing is not yet available. No such attempt has ever been reported. Therefore, at present, we can only be satisfied with a qualitative, or at most a semiquantitative, description of the phenomenon (Marburger, 1975; Shen, 1975). This chapter reviews the theory of SPM and associated spectral broadening. In the following section, we first discuss briefly the various physical mechanisms that can give rise to laser-induced refractive index changes responsible for SPM. Then in Section 3 we present the simple physical picture and theory of SPM and the associated spectral broadening. SPM in space is considered only briefly. Section 4 deals with a more rigorous theory of SPM that takes into account the higher-order effects of the induced refractive index change. Finally, in Section 5, a qualitative picture of how self-focusing can influence

4

Y.R. Shen and G.-Z. Yang

and enhance SPM and spectral broadening is presented. Some semiquantitative estimates of the spectral broadening are given and compared with experiments, including the recent observations of supercontinuum generation in gases.

2. Optical-Field-Induced Refractive Indices The material response to an applied laser field is often nonlinear. An explicit expression for the response is not readily available in general. Unless specified otherwise, we consider here only the case where the perturbative expansion in terms of the applied field is valid and the nonlocal response can be neglected. We can then express the induced polarization in a medium as (Shen, 1984) P(t) = P (1) (t) + P ( 2) (t) + P (3) (t) + L, P (1) (t) = Ú c (1) (t - t ¢) ◊ E(t ¢)dt ¢ = Ú c (1) (w ) ◊ E(w )dw , P ( n) (t) = Ú c ( n) (t - t1, . . . , t - tn ) : E(t1 ) . . . E(tn )dt1 . . . dtn = Ú c ( n) (w = w 1 + w 2 + L + w n ) : E(w 1 ) . . . E(w n )dw 1 . . . dw n ,

(1)

where the applied field is E(t) = Ú E(w )dw

with E(w ) µ exp(-iwt)

(2)

and the nth-order susceptibility is c ( n) (t - t1, . . . ,t - tn ) = Ú c ( n) (w = w 1 + L + w n ) exp[iw 1 (t - t1 ) + L + iw n (t - tn )]dw 1 . . . dw n .

(3)

We note that, strictly speaking, only for a set of monochromatic applied fields can we write P ( n) = c ( n) (w = w 1 + L + w n ) : E(w 1 ) . . . E (w n ).

(4)

In the case of instantaneous response (corresponding to a dispersionless medium), we have P ( n) = c ( n) : [E (t)] . n

(5)

Here, we are interested in the third-order nonlinearity that gives rise to the induced refractive index change. We consider only the self-induced refractive index change; extension to the cross-field-induced change should be straightforward. Thus we assume a pulsed quasi-monochromatic field Ew(t) =

1. Theory of Self-Phase Modulation and Spectral Broadening

5

E (t)exp(-iwt). The third-order nonlinear polarization in a medium, in general, takes the form Pw(3) (t) = Ú Dc (t - t ¢) ◊ E w (t ¢)dt ¢

(6)

with Dc(t - t¢) = Úc(3)(t - t¢, t - t≤, t - t¢≤) : Ew(t≤)E* w(t¢≤)dt≤ dt¢≤. If the optical field is sufficiently far from resonances that the transverse excitations are all virtual and can be considered as instantaneous, we can write Pw(3) (t) = Dc (t) ◊ E w (t), Dc (t) = Ú c (3) (t - t ¢) : E w (t ¢) dt ¢. 2

(7)

In the dispersionless limit, the latter becomes Dc (t) = c (3) : Ew (t) . 2

(8)

Equation (8) is a good approximation when the dispersion of Dc is negligible within the bandwidth of the field. The optical-field-induced refractive index can be defined as Dn = (2p n0 )Dc ,

(9)

where n0 is the average linear refractive index of the medium. With Dn ∫ n2|Ew|2, we have n2 = (2p/n0)c(3). A number of physical mechanisms can give rise to Dc or Dn (Shen, 1966). They have very different response times and different degrees of importance in different media. We discuss them separately in the following.

2.1 Electronic Mechanism Classically, one can imagine that an applied optical field can distort the electronic distribution in a medium and hence induce a refractive index change. Quantum mechanically, the field can mix the electronic wave functions, shift the energy levels, and redistribute the population; all of these can contribute to the induced refractive index change. For a typical transparent liquid or solid, n2 falls in the range between 10-13 and 10-15 esu. For gases at 1 atm pressure, n2 ~ 10-16 to 10-18 esu far away from resonances. The response time is of the order of the inverse bandwidth of the major absorption band (~10-14 to 10-15 s in condensed media) except for the population redistribution part. As the optical frequency approaches an absorption band, n2 is resonantly enhanced. In particular, when the population redistribution due to resonant excitation is significant, the enhancement of n2 can be very large, but the time response will then be dominated by the relaxation of the population redistribution. In a strong laser field, saturation in population redistribution and multiphoton resonant excitations can become important. The perturbative expansion in Eq. (1) may then cease to be valid. For our discussion of SPM

6

Y.R. Shen and G.-Z. Yang

in this chapter, we shall assume that the laser beam is deep in the transparent region and therefore all these electronic resonance effects on the induced refractive index are negligible.

2.2 Vibrational Contribution The optical field can also mix the vibrational wave functions, shift the vibrational levels, and redistribute the populations in the vibrational levels. The corresponding induced refractive index change Dn is, however, many orders (~5) of magnitude smaller than that from the electronic contribution because of the much weaker vibrational transitions. Therefore, the vibrational contribution to Dn is important only for infrared laser beams close to vibrational resonances. For our discussion of SPM, we shall not consider such cases. If the laser pulse is very short (10 fs corresponding to a bandwidth of 500 cm-1), the vibrational contribution to Dn can also come in via Raman excitations of modes in the few hundred cm-1 range. The Raman transitions are also much weaker than the two-photon electronic transitions, so their contributions to the self-induced Dn are usually not important for the discussion of SPM unless femtosecond pulses are used.

2.3 Rotation, Libration, and Reorientation of Molecules Raman excitations of molecular rotations can, however, contribute effectively to Dn. This is because the rotational frequencies of molecules are usually in the few cm-1 region except for the smaller molecules. Thus, even with a monochromatic field, one can visualize a Raman process (in which absorption and emission are at the same frequency w) that is nearly resonant. (The difference frequency of absorption and emission is zero, but it is only a few cm-1 away from the rotational frequencies.) In condensed media, the rotational motion of molecules is, however, strongly impeded by the presence of neighboring molecules. Instead of simple rotations, the molecules may now librate in a potential well set up by the neighboring molecules. The librational frequencies determined by the potential well are often in the range of a few tens of cm-1. The modes are usually heavily damped. Like the rotational modes, they can also contribute effectively to Dn via the Raman process. Molecules can also be reoriented by an optical field against rotational diffusion. This can be treated as an overdamped librational motion driven by the optical field. More explicitly, molecular reorientation arises because the field induces a dipole on each molecule and the molecules must then reorient themselves to minimize the energy of the system in the new environment. All the above mechanisms involving rotations of molecules can contribute appreciably to Dn if the molecules are highly anisotropic. Typically, in liquids, n2 from such mechanisms falls in the range between 10-13 and 10-11 esu, with

1. Theory of Self-Phase Modulation and Spectral Broadening

7

a response time around 10-11 s for molecular reorientation and ~10-13 s for libration. In liquid crystals, because of the correlated molecular motion, n2 can be much larger, approaching 0.1 to 1, but the response time is much longer, of the order of 1 s. The rotational motion is usually frozen in solids, and therefore its contribution to Dn in solids can be neglected.

2.4 Electrostriction, Molecular Redistribution, and Molecular Collisions It is well known that the application of a dc or optical field to a local region in a medium will increase the density of the medium in that region. This is because the molecules in the medium must squeeze closer together to minimize the free energy of the system in the new environment. The effect is known as electrostriction. The induced density variation Dr obeys the driven acoustic wave equation, and from Dn = (∂n/∂r)Dr the induced refractive index change can be deduced. For liquids, we normally have n2 ~ 10-11 esu with a response time of the order of 100 ns across a transverse beam dimension of ~1 mm. Molecules will also locally rearrange themselves in a field to minimize the energy of induced dipole–induced dipole interaction between molecules in the system. Whereas electrostriction yields an isotropic Dn, this molecular redistribution mechanism will lead to an anisotropic Dn. Molecular correlation and collisions could also affect molecular redistribution. A rigorous theory of molecular redistribution is therefore extremely difficult (Hellwarth, 1970). Experimentally, molecular redistribution is responsible for the anisotropic Dn observed in liquids composed of nearly spherical molecules or atoms in cases where the electronic, electrostrictive, and rotational contributions should all be negligible. It yields an n2 of the order of 10-13 esu with a response time in the subpicosecond range. In solids, the molecular motion is more or less frozen, so the contribution of molecular redistribution to Dn is not significant.

2.5 Other Mechanisms A number of other possible mechanisms can contribute to Dn. We have, for instance, laser heating, which increases the temperature of a medium and hence its refractive index; photorefraction, which comes from excitation and redistribution of charged carriers in a medium; and induced concentration variation in a mixture. We conclude this section by noting that there is an intimate connection between third-order nonlinearities and light scattering (Hellwarth, 1977): each physical mechanism that contributes to Dn (except the electronic mechanism) is also responsible for a certain type of light scattering. The thirdorder susceptibility from a given mechanism is directly proportional to the

8

Y.R. Shen and G.-Z. Yang

Table 1.1.

Physical mechanism Electronic contribution Molecular reorientation Molecular libration and redistribution Electrostriction

Magnitude of third-order nonlinearity n2 (esu)

Response time t (s)

10-15-10-13 10-13-10-11 ~10-13

10-14-10-15 ~10-11 ~10-13

~10-11

~10-16*

* For a beam radius of ~1 mm.

scattering cross section related to the same mechanism, and the response time is inversely proportional to the linewidth of the scattering mode. Thus from the low-frequency light scattering spectrum, one can predict the value of n2 for the induced refractive index. For example, in most liquids, light scattering shows a Rayleigh wing spectrum with a broad background extending to a few tens of cm-1. This broad background is believed to arise from molecular libration, redistribution, and collisions (Febellinski, 1967), but the details have not yet been resolved. For our semiquantitative prediction of n2 and the response time, however, we do not really need to know the details if the Rayleigh wing spectrum of the medium is available. A broad and strong Rayleigh wing spectrum is expected to yield a large n2 with a fast response. In Table 1.1 we summarize the results of our discussion of the various physical mechanisms contributing to Dn. It is seen that in nonabsorbing liquid, where all the mechanisms could operate, electrostriction and molecular reorientation may dominate if the laser pulses are longer than 100 ns; molecular reorientation, redistribution, and libration may dominate for pulses shorter than 100 ns and longer than 1 ps; molecular redistribution and libration and electronic contribution may dominate for femtosecond pulses. In transparent solids, usually only electrostriction and electronic contribution are important. Then for short pulses the latter is the only mechanism contributing to Dn.

3. Simple Theory of Self-Phase Modulation and Spectral Broadening For our discussion of SPM of light, let us first consider the case where the propagation of a laser pulse in an isotropic medium can be described by the wave equation of a plane wave: 4p ∂ 2 (3) Ê ∂ 2 n02 ∂ 2 ˆ P , Á 2 - 2 2 ˜E = 2 Ë∂z c ∂t ¯ c ∂t2 where

(10)

1. Theory of Self-Phase Modulation and Spectral Broadening

9

E = E (z,t) exp(ik0 z - iw 0t), P

(3 )

= c (3 ) E E , 2

and n0 is the linear refractive index of the medium. In the simple theory of SPM (Cheung et al., 1968; Gustafson et al., 1969; Shimizu, 1967), we use the usual slowly varying amplitude approximation by neglecting the ∂ 2E /∂t2 term on the left and keeping only the (4p/c2)c(3)|E |2E term on the right of Eq. (10), which then becomes 4pw 02 (3) 2 Ê ∂ n0 ∂ ˆ + c E E. Á ˜E = Ë ∂ z c ∂t ¯ i 2 k0 c 2

(11)

The approximation here also assumes an instantaneous response of c(3). Letting z¢ ∫ z + ct/n0 and E = |E |exp(if), we obtain from the above equation ∂E = 0, ∂ z¢ ∂f 2pw 02 (3) 2 = c E . ∂ z ¢ c 2 k0

(12)

They yield immediately the solution E = E (t) , f (z,t) = f 0 +

(13a)

2 0

2pw 2 c (3) E (t) z. c 2 k0

(13b)

Equation (13a) implies that the laser pulse propagates in the medium without any distortion of the pulse shape, while Eq. (13b) shows that the induced phase change Df(t) = f(z, t) - f0 is simply the additional phase shift experienced by the wave in its propagation from 0 to z due to the presence of the z induced refractive index Dn = (2p/n0)c(3)|E |2, namely Df = (w/c)Ú0 Dn dz. Since the frequency of the wave is w = w0(∂Df/∂t), the phase modulation Df(t) leads to a frequency modulation Dw (t) = - ∂ (Df ) ∂ t 2

=-

2pw 02 (3) ∂ E c z. c 2 k0 ∂t

(14)

The spectrum of the self-phase-modulated field is, therefore, expected to be broadened. It can be calculated from the Fourier transformation E (w ) = 2

1 2p

Ú

•

-•

2

E (t)e - iw t +iwt dt . 0

(15)

An example is shown in Figure 1.1. We assume here a 4.5-ps full width at half-maximum (FWHM) Gaussian laser pulse propagating in a nonlinear medium that yields an SPM output with a maximum phase modulation of

10

Y.R. Shen and G.-Z. Yang

Figure 1.1. Theoretical power spectrum obtained by assuming an instantaneous response of Dn to the intensity variation |E(t)|2, so that the phase modulation Df(t) is proportional to |E(t)|2. (a) Df versus t and (b) power spectrum of the phasemodulated pulse.

Dfmax ⯝ 72p rad. The spectrum of the output shows a broadening of several hundred cm-1 with a quasi-periodic oscillation. It is symmetric with respect to the incoming laser frequency because the SPM pulse is symmetric. The leading half of the Df pulse is responsible for the Stokes broadening and the lagging half for the anti-Stokes broadening. The structure of the spectrum can be understood roughly as follows. As shown in Figure 1.1, the Df curve following the laser pulse takes on a bell shape. For each point on such a curve, one can always find another point with the same slope, except, of course, the inflection points. Since ∂f/∂t = -w, these two points describe radiated waves of the same frequency but different phases. These two waves will interfere with each other. They interfere constructively if the phase difference Df12 is an integer of 2p and destructively if Df12 is an odd integer of p. Such interference then gives rise to the peaks and valleys in the spectrum. The inflection points that have the largest slope on the curve naturally lead to the two outermost peaks with |wmax| ~ |∂f/∂t|max. To find how many peaks we should expect in the spectrum, we need only to know fmax, as the number of pairs of constructive and destructive interferences is simply N ~ fmax/2p on each side of the spectrum. The broadened spectrum has Stokes– anti-Stokes symmetry because Df(t) is directly proportional to |E(t)|2 and is a symmetric pulse.

1. Theory of Self-Phase Modulation and Spectral Broadening

11

Figure 1.2. Theoretical power spectrum obtained by assuming a transient response of Dn to the intensity variation |E(t)|2 so that Df(t) is no longer proportional to |E(t)|2. (a) Df versus t and (b) power spectrum of the phase-modulated pulse.

With the above qualitative picture in mind, we can now generalize our discussion of SPM somewhat. The response of the medium to the laser pulse is generally not instantaneous. One therefore expects z

Df (z,t) = (w c)Ú Dn(z,t)dz, 0

z

Dn(z,t) = Ú n2 (z,t - t ¢) E (z,t ¢) dt ¢. -•

2

(16)

Then, even if |E(t)|2 is symmetric, Df(t) is asymmetric and is no longer proportional to |E(t)|2. The consequence is a Stokes–anti-Stokes asymmetry. An example is given in Figure 1.2. Because of the finite response time of the medium, the leading part of the Df(t) curve always sees a larger portion of the intensity pulse |E(t)|2, and therefore the Stokes side of the spectrum is always stronger. This Stokes–anti-Stokes asymmetry can be drastic if the response time becomes comparable to or smaller than the laser pulse width. In the more rigorous theory, one should also expect a distortion of the pulse shape as the pulse propagates on in the nonlinear medium. Selfsteepening of the pulse, for example, is possible and may also affect the spectral broadening (DeMartini et al., 1967; Gustafson et al., 1969; see Chapter 6). However, the above qualitative discussion still applies since the Df(t) curve should still take on an asymmetric bell shape in general.

12

Y.R. Shen and G.-Z. Yang

The experimental situation is usually not as ideal as the simple theory describes. The laser beam has a finite cross section and will diffract. The transverse intensity variation also leads to a Dn(r) that varies in the transverse directions. This causes self-focusing of the beam and complicates the simultaneously occurring SPM of the beam (Shen, 1975; Marburger, 1975). Moreover, stimulated light scattering could also occur simultaneously in the medium, in most cases initiated by self-focusing (Shen, 1975; Marburger, 1975). All these make the analysis of SPM extremely difficult. One experimental case is, however, close to ideal, namely SPM of a laser pulse in an optical fiber. The transverse beam profile of a guided wave remains unchanged along the fiber. As long as the laser intensity is not too strong, self-focusing and stimulated scattering of light in the fiber can be neglected. For a sufficiently short pulse, the nonlinearity of the fiber is dominated by the electronic contribution and therefore has a nearly instantaneous response. Then if the pulse is not too short and the spectral broadening is not excessive, the slowly varying amplitude approximation is valid and ∂ 2P (3)/∂t2 in the wave equation can be well approximated by -w 20P (3). The only modification of the simple theory of SPM we have discussed is to take into account the fact that we now have a wave in a waveguide with a confined transverse dimension instead of an infinite plane wave in an open space. Thus the quantitative analysis can easily be worked out. Indeed, Stolen and Lin (1978) found excellent agreement between theory and experiment. The above discussion of SPM in time can also be used to describe SPM in space. As we already mentioned, the transverse intensity variation of a laser beam can induce a spatial variation of Dn in the transverse directions. Let us consider here, for simplicity, a continuous-wave (cw) laser beam with a Gaussian transverse profile. The phase increment Df(r, z) varying with the transverse coordinate r is given by Df (r, z) = (w c)Dn(r)z.

(17)

This leads to a distortion of the wave front. Since the beam energy should propagate along the ray path perpendicular to the wave front, this distortion of the wave front would cause the beam to self-focus. If the propagation length is sufficiently long, the beam will actually self-focus and drastically modify the beam cross section. However, if the length of the medium is much shorter than the self-focusing distance, then the self-focusing effect in the medium can be neglected and we are left with only the SPM effect on the beam. The results of Figure 1.1 can describe the spatial SPM equally well if we simply replace t by r and w by k⬜, where k⬜ is the transverse component of the wave vector of the beam. We realize that k⬜ defines the deflection angle q of a beam by the relation k⬜ = (wn0/c)sin q. Therefore, the quasi-periodic spectrum in the k⬜ space actually corresponds to a diffraction pattern with multiple bright and dark rings. This has indeed been observed experimentally (Durbin et al., 1981; Santamato and Shen, 1984). An example is shown in Figure 1.3. Self-focusing or diffraction in the medium can modify the spatial

1. Theory of Self-Phase Modulation and Spectral Broadening

13

Figure 1.3. Diffraction ring pattern arising from spatial self-phase modulation of a CW Ar+ laser beam passing through a 300-mm nematic liquid crystal film. (After Durbin et al., 1981; Santamato and Shen, 1984.)

14

Y.R. Shen and G.-Z. Yang

SPM through its modification of the beam profile. This is analogous to the self-steepening effect on the temporal SPM through its modification of the pulse shape. We now return to the discussion of temporal SPM and spectral broadening. In the next section we consider the case where the incoming laser pulse is very short and spectral broadening is very extensive so that the approximations used in the simple theory of SPM need improvement.

4. More Rigorous Theory of Self-Phase Modulation and Spectral Superbroadening Another experimental case of SPM that could avoid complications arising from self-focusing, stimulated scattering, or other nonlinear optical effects involves the propagation of an ultrashort laser pulse through a thin nonlinear medium. In this case, the medium is thin enough so that the self-focusing effect on SPM in the medium can be ignored. The pulse is short enough so that the transient stimulated light-scattering processes are effectively suppressed. Yet the pulse intensity can still be so high as to induce a very strong SPM, but not high enough to result in appreciable multiphoton absorption or optical breakdown. This is the case first studied by Fork et al. (1987). Using an 80-fs pulse at 627 nm focused to an intensity of 1013 to 1014 W/cm on a 500-mm film of ethylene glycol, they observed in the output a huge spectral broadening that appears as a white continuum. Unlike the spectral broadening discussed in the previous section, the present case shows a Stokes–anti-Stokes asymmetry that emphasizes the anti-Stokes side instead. Such a spectral super-broadening was observed earlier by Alfano and Shapiro (1970) in much longer media with picosecond pulses, but SPM in those cases was definitely affected by self-focusing. Obviously, the results of Fork et al. cannot be explained by the simple theory of SPM. We must resort to a more rigorous analysis. We first notice that the self-steepening effect on the pulse is not included in the simple theory. This means that the approximations neglecting the ∂ 2E /∂t2 term and the terms involving the time derivatives of |P(3)| in the wave equation are not quite appropriate. They become worse for shorter pulses. In the more rigorous analysis of SPM, we must improve on these approximations. Let us now go back to Eq. (10). Without any approximation, we can transform it into an equation for the field amplitude (Yang and Shen, 1984): 1 Ê ∂ n0 ∂ ˆ + Á ˜E + Ë ∂ z c ∂t ¯ i 2 k0 =-

Ê ∂ 2 n02 ∂ 2 ˆ Á 2 - 2 2 ˜E Ë∂z c ∂t ¯

4pw 02 (3) 2 2p Ê ∂ ∂ 2 ˆ (3 ) 2 c E E + i 2 w + Á ˜c E E. 0 i 2 k0 c 2 ik0c 2 Ë ∂t ∂t2 ¯

(18)

1. Theory of Self-Phase Modulation and Spectral Broadening

15

The last term on both sides of the equation has been neglected in the simple theory of SPM. By defining the differential operators D± ∫ (∂/∂z) ± (n0/c)(∂/∂t), Eq. (18) can be written as D+E +

1 1 D- D+E = P, i 2 k0 i 2 k0 4pw 02 È 2i ∂ 1 ∂2 ˘ 2 P = - 2 Í1 + - 2 2 ˙ c (3 ) E E . c Î w0 ∂t w0 ∂t ˚

(19)

Since D- = -(2n0/c)(∂/∂t) + D+, we have from Eq. (19) D+E =

1 i 2 k0

ÈÊ 2n0 ∂ ˆ ˘ ÍÎÁË - c ∂ t + D+ ˜¯ D+E + P ˙˚

=

1 i 2 k0

ÈÊ 2n0 ∂ ˆ ˘ + D+ ˜ i 2k0 ˙ P. ¯ c ∂t ˚ m =0

•

m

 ÍÎÁË -

(20)

It is then simply a question of how many terms in the power series expansion we need to include to better describe the SPM. The zeroth-order approximation corresponds to neglecting all derivatives of c(3)|E |2E any yields D+E = -(2pw 02 ic 2 k0 ) c (3) E E , 2

(21)

which is identical to Eq. (11) used as the basis for the simple theory of SPM. We recognize that under this lowest-order approximation,

(

) (

D+ c (3) E E µ c (3) E 2

<< c

(3 )

)E

2 2 2

E E

if c (3) E

2

<< 1.

(22)

Therefore, we can use D+ as an expansion parameter in the higher-order calculations. For the first-order approximation, we neglect terms involving D +m( c(3)|E |2E ) with m ≥ 1 in Eq. (20) and obtain D+E =

1 i 2 k0

ÈÊ 1 ∂ ˆ m ˘ ˜ P˙ 0 ∂t ¯ m =0 Î ˚ •

 ÍÁË iw

i ∂ ˆ È 2pw 02 (3) 2 ˘ Ê = Á1 + c E E ˙. ˜ Ë w 0 ∂ t ¯ ÍÎ ik0c ˚

(23)

The calculation here has in a sense used c(3)|E |2 as the expansion parameter. In the above first-order approximation, we have kept the ( c(3)|E |2)nE terms with n £ 1 including all their time derivatives. In ordinary cases, this is a very good approximation because usually c(3)|E |2 << 1 and therefore the higherorder terms involving (c(3)|E |2)nE with n ≥ 2 are not very significant. For example, in the ultrashort pulse case, we have c(3) ~ 10-14 esu (or n2 ~ 10-13 esu) for a condensed medium; even if the laser pulse intensity is I ~ 1014 W/cm2, we find c(3)|E |2 ~ 4 ¥ 103 << 1. For larger c(3)|E |2, one may need to include

16

Y.R. Shen and G.-Z. Yang

higher-order terms in the calculations. The next-order correction includes the D+(c(3)|E |2E ) term and all its time derivatives. They yield additional terms proportional to ( c(3)|E |2)2E in the wave equation. If c(3)|E |2 ⲏ 1, then the approach with series expansion will not be useful and we have to go back to the original nonlinear wave equation (19). In the following discussion, we consider only cases with c(3)|E |2 << 1. We are therefore interested in the solution of Eq. (23), which, with n2 = (2pw 0/k0c)c(3), takes the form

(

∂E 1 ∂ + n0E + n2 E ∂ z c ∂t

2

) = i n cw 2

0

2

E E.

(24)

For simplicity, we now neglect the dispersion of the response of the medium. This, as we mentioned earlier, is equivalent to assuming an instantaneous response. Insertion of E = |E |exp(if) into Eq. (24) yields two separate equations for the amplitude and phase: È ∂ n0 Ê 3n2 2 ˆ ∂ ˘ ÍÎ ∂ z + c Ë1 + n E ¯ ∂ t ˙˚ E = 0, 0

(25a)

n 2w 0 2 È ∂ n0 Ê n2 2 ˆ ∂ ˘ ÍÎ ∂ z + c Ë1 + n E ¯ ∂ t ˙˚f = c E . 0

(25b)

In comparison with Eq. (12) for the simple theory of SPM, the only difference is the addition of the n2|E |2 (= Dn) terms on the left-hand sides of Eqs. (25). Its effect is obvious in causing a pulse shape deformation during the pulse propagation. With Dn > 0, we expect a pulse steepening in the lagging edge. This is because the peak of the pulse then propagates at a lower velocity than either the leading or the lagging part of the pulse (DeMartini et al., 1967). Let us first neglect the self-steepening effect on the amplitude pulse. Clearly, self-steepening in the lagging part of the f pulse should lead to a spectral broadening with Stokes–anti-Stokes asymmetry emphasizing the anti-Stokes side, because it is the lagging part of the phase modulation that gives rise to the broadening on the anti-Stokes side. To be more quantitative, we assume an input laser pulse with |E (0, t)|2 = A2/cosh(t/t), whose shape remains unchanged in propagating through the medium so that |E (z, t)|2 = A2/cosh[(t - n0/c)/t]. The solution of Eq. (25b) can then be found analytically as f = w 0t {x - sinh -1[sinh x - (n2 ct )A 2 z]}

(26)

with x = [t - (n0/c)z]/t. The corresponding frequency modulation is given by Dw w 0 = [1 + (Q 2 - 2Q sinh x) cosh x]

-1 2

- 1.

(27)

Here we have defined Q = n2 A 2 z ct

(28)

as a characteristic parameter for spectral broadening. For Q << 1, we have

1. Theory of Self-Phase Modulation and Spectral Broadening

17

Figure 1.4. Maximum Stokes (Dw- < 0) and anti-Stokes (Dw+ > 0) shifts calculated with different models: simple theory of self-phase modulation (---); more rigorous theory without the self-steepening effect on the intensity pulse (–·–); more rigorous theory with the self-steepening effect (—). (After Yang and Shen, 1984.)

Dw w 0 ⯝ - Q sinh x cosh 2 x,

(29)

which is identical to the result one would find from the simple theory of SPM. For Q ⲏ 1, we expect spectral superbroadening with appreciable Stokes–antiStokes asymmetry and a maximum anti-Stokes shift Dw+ ⲏ w 0. The maximum Stokes and anti-Stokes shifts, Dw- and Dw+, respectively, can be directly obtained from Eq. (27): Dw ± w 0 =

1 2

[(Q

2

+ 4)

12

]

± Q - 1.

(30)

This is plotted in Figure 1.4 in comparison with the result calculated from the simple theory of SPM. For |Q| << 1 we have Dw ± ⯝ ±(1/2)w 0|Q|, and for |Q| >> 1 we have Dw + ⯝ w 0|Q| and Dw - ⯝ w 0(1/|Q| - 1). It is seen that Stokes broadening is always limited by |Dw-| < w 0, as it should be. We now include the effect of self-steepening on the amplitude pulse. It can be shown that with |E (0, t)|2 = A2/cosh(t/t), the solution of Eq. (25a) must satisfy the implicit algebraic equation

18

Y.R. Shen and G.-Z. Yang

Figure 1.5. Self-steepening effect on the intensity pulse during its propagation in a nonlinear medium at various values of Q = n2A2zc/t, with x = (t - n0z/c)/t. (After Yang and Shen, 1984.)

[

E (z,t) = A 2 cosh x - Q E (z,t) 2

2

]

A2 .

(31)

A simple numerical calculation then allows us to find |E |2 as a function of x for a given Q. The results are shown in Figure 1.5. For |Q| ⲏ 1, the self-steepening effect is apparent. Knowing |E (z, t)|2, we can again solve for f(z, t) from Eq. (25b) and find Dw(z, t) and Dw±. This can be done numerically; the results are also presented in Figure 1.4. It is seen that for Q > 1, self-steepening of the pulse amplitude has increased the spectral broadening on the anti-Stokes side quite significantly. The additional spectral broadening comes in because the steepening of the amplitude pulse enhances the steepening of the f pulse. The spectral broadening actually results from frequency chirping since ∂f(t)/∂t = -w(t). This is shown in Figure 1.6 for the numerical example discussed above. As expected, the Stokes and anti-Stokes shifts appear, respectively, in the leading and lagging parts of the self-steepened pulse. The Dw = 0 point appears at larger x for larger values of Q because selfsteepening shifts the peak of the pulse to larger x (see Figure 1.5).

1. Theory of Self-Phase Modulation and Spectral Broadening

19

Figure 1.6. Frequency shift due to phase modulation as a function of x = (t - n0z/c)/t at various value of Q = n2A2zc/t.

We can compare the calculation with the experiment of Fork et al. In their experiment, the relevant parameters are n2 ⯝ 10-13 esu, z = 0.05 cm, I ~ 1014 W/cm2, and t (pulse width) ⯝ 8 ¥ 10-14 s. The corresponding value of Q is about 2.3. They observed a Stokes broadening Dw - /w 0 ⯝ -0.6 and an antiStokes broadening Dw+ /w 0 ⯝ 2.3. Our calculation gives Dw- /w 0 ⯝ -0.54 and Dw+/w0 ⯝ 3.5. Considering the uncertainty in the experimental parameters, we can regard the agreement between theory and experiment as reasonable. Manassah et al. (1985, 1986) used the method of multiple scales to solve Eq. (18) (neglecting dispersion). They also took n2|E |2/n0 as the expansion parameter in their series expansion and therefore necessarily obtained the same results as we discussed above.* We have neglected in the above calculation the dispersion of the medium response. Normal dispersion may also reshape the pulse (Fisher and Bischel, 1975; Fisher et al., 1983), but in the present case the length of the medium is so short that this effect is not likely to be important. Anomalous dispersion with resonances in c(3) or n2 could, however, give rise to resonant structure in the broadened spectrum. The calculation including the dispersion of c(3) is much more complicated and, in general, must resort to numerical solution (Fisher and Bischel, 1975; Fisher et al., 1983). In obtaining the timedependent solution of the wave equation with the third-order nonlinearity, we have already taken all the four-wave mixing contributions into account. By adding a noise term with a blackbody spectrum in the nonlinear wave equation, the four-wave parametric generation process proposed by Penzkofer et al. (1973, 1975) could also be inclued in the calculation. * A factor of 3 in front of n2 is mistakenly left out in Eq. (3a) of Yang and Shen (1984).

20

Y.R. Shen and G.-Z. Yang

5. Self-Focusing and Self-Phase Modulation For pulsed laser beam propagation in a nonlinear medium, SPM in time and SPM in space necessarily appear together. SPM in time causes selfsteepening of the pulse, which in turn enhances SPM in time. Similarly, SPM in the transverse beam profile causes self-focusing of the beam, which in turn enhances the transverse SPM. If the propagation distance in the medium is sufficiently long, these effects can build up to a catastrophic stage, namely self-steepening to a shock front and self-focusing to a spot limited in dimensions only by higher-order nonlinear processes and diffraction. SPM in time and SPM in space are then tightly coupled and strongly influenced by each other. In many experiments, the observed strong temporal SPM and extensive spectral broadening are actually initiated by self-focusing. In such cases, the input laser pulse is so weak that without self-focusing in the nonlinear medium, SPM would not be very significant. Self-focusing to a limiting diameter greatly enhances the beam intensity, and hence SPM can appear several orders of magnitude stronger. A quantitative description of such cases is unfortunately very difficult, mainly because the quantitative theory for selffocusing is not yet available. We must therefore restrict ourselves to a more qualitative discussion of the problem.

5.1 Self-Phase Modulation with Quasi-Steady-State Self-Focusing In the early experiments on self-focusing of single-mode nanosecond laser pulses, it was found that the output of the self-focused light had a spectral broadening of several hundred cm-1 (Shen, 1975; Marburger, 1975). This was rather surprising because from the simple theory of SPM, picosecond pulses would be needed to create such a spectral broadening (Cheung et al., 1968; Gustafson et al., 1969; Shimizu, 1967). Later, the observation was explained by SPM of light trailing behind a moving focus (Shen and Loy, 1971; Wong and Shen, 1972). We briefly review the picture here and then use it to interpret the recently observed SPM and spectral superbroadening of ultrashort pulses in gases (Corkum et al., 1986; Glownia et al., 1986). Figure 1.7 depicts the quasi-steady-state self-focusing of a laser pulse leading to a moving focus along the axis (zˆ) with a U-shaped trajectory described by the equation (Shen, 1975; Marburger, 1975) zf (t) =

K , P (tR ) - P0

(32)

where P0 is the critical power for self-focusing, P (tR ) is the laser power at the retarded time tR = t - zf n0/c, and K is a constant that can be determined from experiment. This equation assumes instantaneous response of Dn to the applied field, which is a good approximation as long as the response time t is much shorter than the laser pulse width. Here, we are interested only in the

1. Theory of Self-Phase Modulation and Spectral Broadening

21

Figure 1.7. Self-focusing for an input laser pulse in (a) leading to the trajectory of a moving focus in the form of U curve in (b). The dashed lines in (b), with slope equal to the light velocity, depict how light propagates in the medium along the z axis at various times. The shaded region around the U curve has appreciably larger Dn. Light traversing the medium along the dashed lines through the shaded region should acquire a phase increment Df that varies with time.

upper branch of the U curve, along which the focus has a forward velocity faster than light. Because of the high laser intensity in the focal spot, the locally induced Dn should be appreciable and should last for a duration not shorter than the relaxation time t. Thus one can imagine that the moving focus creates in the medium a channel of Dn at least t dzf /dt long trailing after the focus. Consider now the defocused light from a local focal spot. Since it lags behind the moving focus (which travels faster than light), it experiences the Dn dielectric channel created by the focus over a certain distance and will diffract only weakly. In other words, the defocused light from the focus is partially trapped in the Dn channel. This partial trapping of light in turn helps to maintain the Dn channel and make it last longer. The emission from the focal spot at the end of the medium then takes the form of an asymmetric pulse (with a pulse width of the order of a few t) with a longer trailing edge. The above picture is also illustrated in Figure 1.7. We use the shaded area around the U curve to denote the region with appreciable Dn. The laser input at tA focuses at A, but defocuses more gradually because of the existing Dn channel in front of it. The partially trapped light then propagates along the axis from A to the end of the medium at A¢, crossing the shaded region with appreciable Dn. It therefore acquires a significant phase increment Df. From the figure, one may visualize that Df can be strongly phase modulated in time, varying from nearly zero to a maximum and back to zero in a few relaxation times. This could yield appreciable spectral broadening in the output of the self-focused light. To be more quantitative, we realize that the light pulse emitted from a focus in the medium must be asymmetric and must have a pulse width of several t. The shaded area in Figure 1.7 has a somewhat larger width since Dn is induced by the focused light. Knowing the trajectory of the moving focus,

22

Y.R. Shen and G.-Z. Yang

the beam intensity in the focal region, and how Dn responds to the intensity, we can calculate Df(t) and hence Dw(t) and the broadened spectrum (Shen and Loy, 1971; Wong and Shen, 1972). As an example, consider the case of a 1.2-ns laser pulse propagating into a 22.5-cm CS2 cell. The trajectory of the moving focus (focal diameter ⯝ 5 mm) is described by Eq. (32) with K = 5.7 (kW)1/2-cm and P0 = 8 kW. In this case, Dn is dominated by molecular reorientation; it obeys the dynamic equation n2 Ê ∂ 1ˆ 2 E (z, t) . Á + ˜ Dn = Ë ∂t t ¯ t

(33)

For CS2, t = 2 ps and n2 = 10-11 esu. The phase increment experienced by light waves traversing the cell along the axis is given by l

Df = Ú (w c)Dn[z, t ¢ = t - (l - z) n0 c]dz, 0

(34)

Figure 1.8. Theoretical power spectrum of a light pulse emitted from the focal region of a moving focus at the end of a CS2 cell. (a) The intensity pulse; (b) Df versus t; (c) the power spectrum (see text for details). (After Shen and Loy, 1971; Wong and Shen, 1972.)

1. Theory of Self-Phase Modulation and Spectral Broadening

23

where l is the cell length. (For this illustrative example, we have neglected the diffraction effect on Df.) We now simply assume that |E(z, t)|2 in the focal region resulting from self-focusing has a pulse width of ~ 3t and a pulse shape as shown in Figure 1.8a. Equations (33) and (34) then allow us to find Dn(z, t) and Df(t). Knowing Df(t) and E(l, t), we can then calculate the spectrum of the output from the focal spot at the end of the cell, as shown in Figure 1.8c. The experimentally observed spectrum has in fact the predicted spectral broadening (Shen and Loy, 1971; Wong and Shen, 1972), but it often has a strong central peak (Figure 1.9). This is presumably because in the

Figure 1.9. (a) Experimentally observed power spectrum of light emitted from the focal region at the end of a 10-cm CS2 cell; the input pulse has a pulse width of 1.2 ns and a peak power of 27 kW. (b) Theoretical power spectrum using the moving focus model. (After Shen and Loy, 1971; Wong and Shen, 1972.)

24

Y.R. Shen and G.-Z. Yang

calculation we have neglected a significant portion of the beam that selffocuses from the periphery and experiences little phase modulation. For shorter input pulses of longer cells, self-focusing of the beam toward the end of the cell is more gradual; accordingly, the weakly phase-modulated part is less and the central peak in the spectrum is reduced. We also note that in Figures 1.8 and 1.9 the anti-Stokes broadening is much weaker. This is because the negatively phase-modulated part of the pulse has little intensity, as seen in Figure 1.8. Using the picture sketched in Figure 1.7, we can actually predict the Stokes broadening with the correct order of magnitude by the following rough estimate. We approximate the upper branch of the U curve toward the end of the medium by a straight line with a slope equal to the end velocity of the moving focus. If Dn is the induced refractive index in the shaded area, then the phase modulation of the emitted light is given by Df (t) ⯝

Ê w 0 ˆ Ê n0 1 ˆ Ë c ¯Ë c v¯

-1

Ú

t

t0

Dn(l ,t ¢)dt ¢,

(35)

where l is the length of the medium and t0 is the time when Dn(l, t) starts to become appreciable. The extent of Stokes broadening is readily obtained from Dw - = (∂ Df ∂ t) max ⯝ w 0Q

(36)

with Q = Dnmaxleff/cT where T ~ 2t and leff = T/[(n0/c) - (1/v)]. For the above example with CS2, we have Dnmax ~ 10-3, T ~ 4 ps, leff ~ 1 cm, and Q ~ 0.01. The resultant Stokes broadening should be Dw- ~ 150 cm-1. The experimentally observed broadening is about 120 cm-1.

5.2 Spectral Superbroadening of Ultrashort Pulses in Gases The above discussion can be used to explain qualitatively the recently observed spectral superbroadening of ultrashort pulses in gas media (Corkum et al., 1986; Glownia et al., 1986). In those experiments, picosecond or femtosecond laser pulses with energies of several hundred microjoules were weakly focused into a high-pressure gas cell. Spectral superbroadening with Dw ~ 104 cm-1 was observed. A few examples are shown in Figure 1.10. Self-focusing was apparently present in the experiment. We therefore use the above simple model for SPM with quasi-steady-state self-focusing to estimate the spectral broadening (Loy and Shen, 1973), assuming that Dn from the electronic contribution in the medium has a response time t ~ 10 fs. In this case, the position of the moving focus is given by (Shen, 1975; Marburger, 1975)

(z -f 1 (t) - f -1 ) -1 =

K P (tR ) - P0

(37)

instead of Eq. (32), where f is the focal length of the external focusing lens. Let us consider, for example, external focusing of a 250-mJ, 100-fs pulse to a

1. Theory of Self-Phase Modulation and Spectral Broadening

25

Figure 1.10. Continuum spectra of self-phase-modulated light from 70-fs pulses in 30-atm xenon (crosses), 2-ps pulses in 15-atm xenon (circles), and 2-ps pulses in 40-atm nitrogen (squares). The cell length is 90 cm. (After Corkum et al., 1986.)

nominally 100-mm focal spot in a 3-atm, 100-cm Xe cell. Self-focusing yields a smaller focus, assumed to be 50 mm. We then use the values n2 ~ 10-16 esu, I ~ 1014 W/cm2, Dnmax ~ 4 ¥ 10-5, leff ~ 10 cm, and T ~ 2t ~ 20 fs; we find Q ~ 1 and hence Dw- ~ 104 cm-1. The above estimate is admittedly very crude because of uncertainties in the experimental parameters, but it does give a spectral superbroadening in order-of-magnitude agreement with the experiments. Appreciable anti-Stokes broadening was also observed in the superbroadened spectrum of the SPM light from a gas medium. This seems to be characteristically different from what we have concluded from the discussion in the previous subsection. However, we realize that in the present case the moving focus terminates at z = f instead of the end of the cell, and the total transmitted light is detected and spectrally analyzed. Thus the detected output pulse has essentially the same intensity envelope as the input pulse if we neglect the self-steepening effect, and the negatively phase-modulated part (the trailing edge) of the Df(t) pulse will overlap with the major part of the intensity pulse. Consequently, the spectral intensity of the anti-Stokes side should be nearly as strong as that of the Stokes side. The extent of the antiStokes broadening is expected to be somewhat less than that of the Stokes broadening because of the longer trailing edge of the Df pulse, unless the self-steepening effect becomes important. Self-focusing in a gas medium should be more gradual than in a liquid cell. With weak external focusing, the focal dimensions resulting from combined external and self-focusing may not be very different from those resulting from external focusing alone. Thus, even with self-focusing, the SPM output from

26

Y.R. Shen and G.-Z. Yang

the gas medium may not have a much larger diffraction angle than the linearly transmitted output, as was observed in the experiments.

5.3 Self-Phase Modulation with Transient Self-Focusing We have used the picture of a moving focus with a trailing dielectric channel to describe SPM initiated by quasi-steady-state self-focusing. For shorter input pulses, the velocity of the forward moving focal spot is closer to the light velocity, and consequently more light is expected to be trapped in the dielectric channel for a longer distance. In fact, when the pulse width is comparable to or shorter than the relaxation time t, the entire self-focusing process becomes transient, and the input pulse will evolve into a dynamic trapping state (Loy and Shen, 1973). The dynamic trapping model for transient self-focusing is an extension of the moving focus model for quasi-steady-state self-focusing. Consider the case where Dn is governed by Eq. (33) or, more explicitly, Dn(z, x ) =

1 x 2 n2 E (z, x ¢) exp[-(x - x ¢) t ] dx ¢, t Ú-•

(38)

where x = t - zn0/c. Because of this transient response of Dn, the later part of the pulse propagating in the medium may see a larger Dn than the earlier part. As a result, different parts of the pulse will propagate in the medium differently, as sketched in Figure 1.11 (Loy and Shen, 1973). The transient Dn makes the very leading edge of the pulse diffract, the middle part self-focus weakly, and the lagging part self-focus to a limiting diameter. The result is that in

Figure 1.11. Sketch showing self-focusing of an ultrashort pulse in a medium with a transient response of Dn. Different parts (a, b, c, etc.) of the pulse focus and defocus along different ray paths. The pulse is first deformed into a horn shape and then propagates on without much further change. (After Loy and Shen, 1973.)

1. Theory of Self-Phase Modulation and Spectral Broadening

27

propagating through the medium, the pulse is first deformed into a hornshaped pulse and then the horn-shaped pulse propagates on with only a slight change of the pulse shape due to diffraction of the front edge. In a long medium, the front-edge diffraction finally could blow up the deformed pulse. Note that this picture comes in because in transient self-focusing, both focusing and diffraction are much more gradual, leading to a long longitudinal focal dimension and hence the rather stable horn-shaped propagating pulse. Such a stable form of self-focused pulse propagation is known as dynamic trapping. Since the various parts of the light pulse see different Dn’s along their paths, the phase increments Df they acquire are also different. This means phase modulation and hence spectral broadening. As an approximation, we can assume that the overall phase modulation is dominated by that of a stable horn-shaped pulse propagating in the nonlinear medium over a finite length leff. For illustration, let us take an example in which the horn-shaped pulse can be described by E (r, x ) = A 02 exp[-x 2t p2 - 2r 2 r 02 (x )] 2

(39)

with r0 (x ) a = 1

if x £ x1

= (1 - D)e

- ( x - x1) t1

+D

= (1 - D)e

- ( x2 - x1) t1

+ D if x ≥ x 2 ,

if x1 £ x £ x 2

where x = t - zn0/c. We have picosecond pulse propagation in Kerr liquids in mind and therefore choose tp = 1.25t, x1 = 2.5t, x2 = 2t, t1 = t, D = 0.05, and t = 2 ps (for CS2). We also choose the pulse intensity as A20 = 80(n2/n0)k2a2, where k is the wave vector and the effective pulse propagation distance leff = 0.15 ka2. From the three-dimensional wave equation, it can be shown that the phase modulation obeys the equation ∂ (Df ) ∂ z = k (Dn n0 ) - 2 kr 02 (x ).

(40)

The second term on the right of the equation is the diffractive contribution to Df, which can be appreciable when r0 is small. Knowing |E(r, x)|2, we can find Dn(z, x) from Eq. (38), and hence Df(z, x) from Eq. (40), and finally the broadened spectrum from |E(r, x)|2, and Df(z, x) for z = leff, as shown in Figure 1.12. The main qualitative result of the above calculation is that the spectrum has the quasi-periodic structure with nearly equal Stokes and anti-Stokes broadening, although the Stokes side is more intense. This agrees with the experimental observation (Cubbedu and Zagara, 1971; Cubbedu et al., 1971) and the more detailed numerical calculation of Shimizu and Courtens (1973). The reason is as follows. The Stokes–anti-Stokes symmetry results from a symmetric Df(t) pulse that overlaps well with the intensity pulse. The neck portion of the horn-shaped pulse with ∂(Df)/∂t < 0 contributes to the antiStokes broadening. As seen in Eq. (40), the time dependence of Df comes

28

Y.R. Shen and G.-Z. Yang

Figure 1.12. Theoretical power spectrum obtained by assuming a horn-shaped pulse propagating for a certain distance in a nonlinear medium without any change in its shape. (a) Normalized intensity output pulse; (b) Df versus t; (c) power spectrum of the output. (After Loy and Shen, 1973.)

solely from the time dependence in (Dn/n0 - 2/k2r20). Without the diffraction term (-1/k2r20), the Df pulse would have a longer trailing edge because of the relaxation of Dn. With the diffraction term, the rapid reduction of r0 toward the neck makes the Df pulse more symmetric. Thus it appears that the dynamic trapping model explains fairly well the qualitative features of SPM and spectral broadening initiated by transient self-focusing. The broadening is more extensive with more intense input pulses and longer propagation lengths. In a long medium, dynamic trapping may exist only over a limit section of the medium. In that case, the spectrally broadened light may seem to have originated from a source inside the medium. This has also been observed experimentally (Ho et al., 1987).

6. Conclusion We have seen that the temporal SPM and the concomitant spectral broadening of light arise because an intense optical field can induce an appreciable refractive index change Dn in a medium. The theory of pure SPM is, in principle, quite straightforward. If the input pulse intensity is not very strong, the zeroth-order approximation taking into account only the direct contribution of Dn to the induced phase change Df should already give a fairly good description. The next-order approximation including the self-steepening effect on both Df and the amplitude pulse should satisfactorily cover the cases of strong SPM with spectral superbroadening.

1. Theory of Self-Phase Modulation and Spectral Broadening

29

Unfortunately, the temporal SPM is often complicated by the spatial SPM. The latter can lead to self-focusing, which dramatically alters the intensity distributions of the laser pulse in space and time and therefore drastically modifies the temporal SPM. In fact, in most practical cases, self-focusing occurs long before the temporal SPM becomes appreciable; it is actually selffocusing that increases the beam intensity in the medium and thus initiates a strong SPM in time. Only by using an optical waveguide or a very thin nonlinear medium can self-focusing be avoided. These are then the only experimental cases where a pure temporal SPM has been realized. In the pure SPM case, the theoretical difficulty is in the description of Dn: it is not easy, in general, to predict quantitatively the nonlinear response of a medium from first principles; one must rely on experimental measurements. Quantitative measurements of Dn(t) in the picosecond and femtosecond domains are still rare. In particular, measurements of Dn with femtosecond time resolution are still rather difficult. The various low-frequency resonances could make the time dependence of Dn very complex. Inclusion of the transient response of Dn (or the dispersion of Dn) in the theory complicates the calculation; one may have to resort to numerical solution of the problem. Experimentally, SPM of laser pulses in optical fibers has been well studied; SPM of ultrashort pulses in thin nonlinear media is, however, still not well documented. More careful quantitative measurements are needed in order to have a more detailed comparison with theory. With self-focusing, the theory of SPM becomes extremely complex. The main difficulty lies in the fact that a quantitative theory capable of describing the details of self-focusing is not yet available. We must then rely on the more qualitative physical pictures for self-focusing. Therefore, the discussion of the subsequently induced SPM can be at most semiquantitative. Thus, we find it quite satisfying that the predicted spectral broadening from such theoretical discussions can give order-of-magnitude agreement with the experimental observations in a number of very different cases: nanosecond pulse propagation in liquids to pico- or femtosecond pulse propagation in gases. Spectral superbroadening is often observed with the propagation of ultrashort pulses in condensed media and is apparently initiated by self-focusing. In most cases, the details of the self-focusing process have not been measured; in some cases, even the quantitative information on Dn is not available. The measurements on spectral broadening also tend to show strong fluctuations. All these made even an order-of-magnitude comparison between theory and experiment rather difficult. A complete theory of SPM with self-focusing requires the solution of a time-dependent three-dimensional wave equation. With self-focusing modifying the laser pulse rapidly in both space and time, such a solution, even on the largest computer, is a formidable task. In our opinion, the best way to tackle the problem is to try to simplify the calculation by reasonable approximations derived from the physical picture that has already been established for self-focusing.

30

Y.R. Shen and G.-Z. Yang

Acknowledgment. This work was supported by the Director, Office of Energy Research, Office of Basic Energy Sciences, Materials Sciences Division of the U.S. Department of Energy under Contract No. DE-AC03-76SF00098.

References Alfano, R.R. and S.L. Shapiro (1970) Emission in the region 4000–7000 A via fourphoton coupling in glass; Observation of self-phase modulation and small scale filaments in crystals and glasses; Direct distortion of electronic clouds of rare-gas atoms in intense electric fields. Phys. Rev. Lett. 24, 584, 592, 1219. Alfano, R.R. and S.L. Shapiro (1971) Picosecond spectroscopy using the inverse Raman effect. Chem. Phys. Lett. 8, 631. Alfano, R.R., Q. Li, T. Jimbo, J.T. Manassah, and P.P. Ho (1986) Induced spectral broadening of a weak picosecond pulse in glass produced by an intense picosecond pulse. Opt. Lett. 11, 626. Alfano, R.R., Q. Wang, T. Jimbo, and P.P. Ho (1987) Induced spectral broadening about a second harmonic generated by an intense primary ultrashort laser pulse in ZuSe. Phys. Rev. A 35, 459. Bloembergen N. and P. Lallemand (1966) Complex intensity-dependent index of refraction, frequency broadening of stimulated Raman lines, and stimulated Rayleigh scattering. Phys. Rev. Lett. 16, 81. Brewer, R.G. (1967) Frequency shifts in self-focusing light. Phys. Rev. Lett. 19, 8. Busch, G.E., R.P. Jones, and P.M. Rentzepis (1973) Picosecond spectroscopy using a picosecond continuum. Chem. Phys. Lett. 18, 178. Cheung, A.C., D.M. Rank, R.Y. Chiao, and C.H. Townes (1968) Phase modulation of Q-switched laser beams in small-scale filaments. Phys. Rev. Lett. 20, 786. Corkum, P.B., P.P. Ho, R.R. Alfano, and J.T. Manassah (1985) Generation of infrared supercontinuum covering 3–14 mm in dielectrics and semiconductors. Opt. Lett. 10, 624. Corkum, P.B., C. Rolland, and T. Rao (1986) Supercontinuum generation in gases. Phys. Rev. Lett. 57, 2268. Cubbedu, R. and F. Zagara (1971) Nonlinear refractive index of CS2 in small-scale filaments. Opt. Commun. 3, 310. Cubbedu, R., R. Polloni, C.A. Sacchi, O. Svelto, and F. Zagara (1971) Study of smallscale filaments of light in CS2 under picosecond excitation. Phys. Rev. Lett. 26, 1009. DeMartini, F., C.H. Townes, T.K. Gustafson, and P.L. Kelley (1967) Self-steepening of light pulses. Phys. Rev. 164, 312. Durbin, S.D., S.M. Arakelian, and Y.R. Shen (1981) Laser-induced diffraction rings from a nematic liquid crystal film. Opt. Lett. 6, 411. Fabellinski, I.L. (1967) Molecular Scattering of Light. Plenum, New York, Chapter VIII. Fisher, R.A. and W. Bischel (1975) Numerical studies of the interplay between selfphase modulation and dispersion for intense plane-wave light pulses. J. App. Phys. 46, 4921. Fisher, R.A., B. Suydam, and D. Yevich (1983) Optical phase conjugation for time domain undoing of dispersive self-phase modulation effects. Opt. Lett. 8, 611. Fork, R.L., C.V. Shank, C. Hirliman, and R. Yen (1983) Femtosecond white-light continuum pulses. Opt. Lett. 8, 1.

1. Theory of Self-Phase Modulation and Spectral Broadening

31

Fork, R.L., C.H. Brito Cruz, P.C. Becker, and C.V. Shank (1987). Compression of optical pulses to six femtoseconds by using cubic phase compensation. Opt. Lett. 12, 483. Glownia, J., G. Arjavalingam, P. Sorokin, and J. Rothenberg (1986) Amplification of 350-fs pulses in XeCl excimer gain modules. Opt. Lett. 11, 79. Gustafson, T.K., J.P. Taran, H.A. Haus, J.R. Lifsitz, and P.L. Kelley (1969) Self-phase modulation, self steepening, and spectral development of light in small-scale trapped filaments. Phys. Rev. 177, 306. Hellwarth, R.W. (1970) Theory of molecular light scattering spectra using the lineardipole approximation. J. Chem. Phys. 52, 2128. Hellwarth, R.W. (1977) Third-order optical susceptibilities of liquids and solids. Prog. Quantum Electron. 5, 1. Ho, P.P., Q.X. Li, T. Jimbo, Y.L. Ku, and R.R. Alfano (1987) Supercontinuum pulse generation and propagation in a liquid CCl4. Appl. Opt. 26, 2700. Ippen, E.P. and C.V. Shank (1975) Dynamic spectroscopy and subpicosecond pulse compression. Appl. Phys. Lett. 27, 488. Jones, W.J. and B.P. Stoicheff (1964) Induced absorption at optical frequencies. Phys. Rev. Lett. 13, 657. Lallemand, P. (1996) Temperature variation of the width of stimulated Raman lines in liquids. Appl. Phys. Lett. 8, 276. Loy, M.M.T. and Y.R. Shen (1973) Study of self-focusing and small-scale filaments of light in nonlinear media. IEEE J. Quantum Electron. QE-9, 409. Manassah, J.T., R.R. Shapiro, and M. Mustafa (1985) Spectral distribution of an ultrashort supercontinuum laser source. Phys. Lett. A107, 305. Manassah, J.T., M.A. Mustafa, R.R. Alfano, and P.P. Ho (1986) Spectral extent and pulse shape of the supercontinuum for ultrashort laser pulse. IEEE J. Quantum Electron. QE-22, 197. Marburger, J.H. (1975) Self-focusing: theory. Prog. Quantum Electron. 4, 35. Nakatsuka, H. and D. Grischkowsky (1981) Recompression of optical pulses broadened by passage through optical fibers. Opt. Lett. 6, 13. Nakatsuka, H., D. Grischkowsky, and A.C. Balant (1981) Nonlinear picosecond pulse propagation through optical fibers with positive group velocity dispersion. Phys. Rev. Lett. 47, 910. Nikolaus, B. and D. Grischkowsky (1983a) 12¥ pulse compression using optical fibers. Appl. Phys. Lett. 42, 1. Nikolaus, B. and D. Grischkowsky (1983b) 90-fs tunable optical pulses obtained by two-state pulse compression. Appl. Phys. Lett. 43, 228. Penzkofer, A. (1974) Parametrically generated spectra and optical breakdown in H2O and NaCl. Opt. Commun. 11, 265. Penzkofer, A., A. Laubereau, and W. Kaiser (1973) Stimulated short-wave radiation due to single-frequency resonances of c(3). Phys. Rev. Lett. 31, 863. Penzkofer, A., A. Seilmeier, and W. Kaiser (1975) Parametric four-photon generation of picosecond light at high conversion efficiency. Opt. Commun. 14, 363. Santamato, E. and Y.R. Shen (1984) Field curvature effect on the diffraction ring pattern of a laser beam dressed by spatial self-phase modulation in a nematic film. Opt. Lett. 9, 564. Shen, Y.R. (1966) Electrostriction, optical Kerr effect, and self-focusing of laser beams. Phys. Lett., 20, 378. Shen Y.R. (1975) Self-focusing: experimental. Prog. Quantum Electron. 4, 1.

32

Y.R. Shen and G.-Z. Yang

Shen, Y.R. (1984) The Principles of Nonlinear Optics. Wiley, New York, Chapters 1 and 16. Shen, Y.R. and M.M.T. Loy (1971) Theoretical investigation of small-scale filaments of light originating from moving focal spots. Phys. Rev. A 3, 2099. Shimizu, F. (1967) Frequency broadening in liquids by a short light pulse. Phys. Rev. Lett. 19, 1097. Shimizu, F. and E. Courtens (1973) Recent results on self-focusing and trapping. In Fundamental and Applied Laser Physics, M.S. Feld, A. Javan, and N.A. Kurnit, eds. Wiley, New York, p. 67. Stoicheff, B.P. (1963) Characteristics of stimulated Raman radiation generated by coherent light. Phys. Lett. 7, 186. Stolen, R. and C. Lin (1978) Self-phase modulation in silica optical fibers. Phys. Rev. A 17, 1448. Topp, M.R. and P.M. Rentzepis (1971) Time-resolved absorption spectroscopy in the 10-12-sec range. J. Appl. Phys. 42, 3415. Treacy, E.P. (1968) Compression of picosecond light pulses. Phys. Lett. 28A, 34. Treacy, E.P. (1969a) Measurements of picosecond pulse substructure using compression techniques. Appl. Phys. Lett. 14, 112. Treacy, E.P. (1969b) Optical pulse compression with diffraction gratings. IEEE J. Quantum Electron. 5, 454. Wong, G.K.L. and Y.R. Shen (1972) Study of spectral broadening in a filament of light. Appl. Phys. Lett. 21, 163. Yang, G. and Y.R. Shen (1984) Spectral broadening of ultrashort pulses in a nonlinear medium. Opt. Lett. 9, 510.

2

Supercontinuum Generation in Condensed Matter Q.Z. Wang, P.P. Ho, and R.R. Alfano

1. Introduction Supercontinuum generation, the production of intense ultrafast broadband “white light” pulses, arises from the propagation of intense picosecond or shorter laser pulses through condensed or gaseous media. Various processes are responsible for continuum generation. These are called self-, induced-, and cross-phase modulations and four-photon parametric generation. Whenever an intense laser pulse propagates through a medium, it changes the refractive index, which in turn changes the phase, amplitude, and frequency of the incident laser pulse. A phase change can cause a frequency sweep within the pulse envelope. This process has been called self-phase modulation (SPM) (Alfano and Shapiro, 1970a). Nondegenerate four-photon parametric generation (FPPG) usually occurs simultaneously with the SPM process (Alfano and Shapiro, 1970a). Photons at the laser frequency parametrically generate photons to be emitted at Stokes and anti-Stokes frequencies in an angular pattern due to the required phase-matching condition. When a coherent vibrational mode is excited by a laser, stimulated Raman scattering (SRS) occurs. SRS is an important process that competes and couples with SPM. The interference between SRS and SPM causes a change in the emission spectrum resulting in stimulated Raman scattering cross-phase modulation (SRSXPM) (Gersten et al., 1980). A process similar to SRS-XPM occurs when an intense laser pulse propagates through a medium possessing a large secondorder c2 and third-order c3 susceptibility. Both second harmonic generation (SHG) and SPM occur and can be coupled together. The interference between SHG and SPM alters the emission spectrum and is called second harmonic generation cross-phase modulation (SHG-XPM) (Alfano et al., 1987). A process closely related to XPM, called induced phase modulation (IPM) (Alfano, 1986), occurs when a weak pulse at a different frequency propagates through a disrupted medium whose index of refraction is changed by an intense laser pulse. The phase of the weak optical field can be modulated by the time variation of the index of refraction originating from the primary intense pulse.

34

Q.Z. Wang, P.P. Ho, and R.R. Alfano

Table 2.1. Brief history of experimental continuum generation.

Investigator

Year

Material

Alfano, Shapiro

1968–1973

Liquids and solids

Stolen et al. Shank, Fork et al.

1974–1976 1983

Fibers Glycerol

Corkum, Ho, Alfano Corkum, Sorokin Alfano, Ho, Manassah, Jimbo Alfano, Ho, Wang, Jimbo Alfano, Ho, Baldeck

1985 1986

Semiconductors dielectrics Gases

1986

Glass

1986

1987

Laser wavelength/ pulsewidth 530 nm/ 8 ps or 1060 nm/ 8 ps 530 nm/ns 620 nm/100 fs

Spectrum

Frequency (cm-1)

Process

Visible and near IR

6,000

SPM

Visible UV, visible, near IR IR

500 10,000

SPM SPM

1,000

SPM

600 nm/2 ps 300 nm/0.5 ps 1,060 nm/ 530 nm/8 ps

Visible and UV Visible

5,000

SPM

1,000

IPM (XPM)

ZnSe

1,060 nm/8 ps

Visible

1,000

SHG-XPM (ISB)

Fibers

530 nm/30 ps

Visible

1,000

SRS-XPM

10 mm/6 ps

The first study of the generation and mechanisms of the ultrafast supercontinuum dates back to the years 1968 to 1972, when Alfano and Shapiro first observed the “white” picosecond continuum in liquids and solids (Alfano and Shapiro, 1970a). Spectra extending over ~6000 cm-1 in the visible and infrared wavelength region were observed. Over the years, improvements in the generation of ultrashort pulses from mode-locked lasers led to the production of wider supercontinua in the visible, ultraviolet, and infrared wavelength regions using various materials. Table 2.1 highlights the major accomplishments in this field over the past 20 years. In this chapter we focus on the picosecond supercontinuum generation in liquids, solids, and crystals. Supercontinuum generation in gases, XPM, and IPM are discussed by Corkum and Rolland (Chapter 7), Glownia et al. (Chapter 8), Baldeck et al. (Chapter 4), Agrawal (Chapter 3), and Manassah (Chapter 5), respectively.

2. Simplified Model Before we go further, let us first examine the nonlinear wave equation to describe the self-phase modulation mechanism. A thorough theoretical study of supercontinuum generation has been dealt with in Chapters 1, 3, and 5.

2. Supercontinuum Generation in Condensed Matter

35

The optical electromagnetic field of a supercontinuum pulse satisfies Maxwell’s equations: 1 ∂B , c ∂t 1 ∂ D 4p —¥H= + J, c ∂t c —◊ D = 4pr, —◊ B = 0. —¥E=-

(1)

Equations (1) can be reduced to (see Appendix) ∂A 1 ∂A w 0 n2 2 + =i A A, ∂z vg ∂ t 2c

(2)

where A(z, t) is the complex envelope of the electric field and v g = 1/(∂ k/∂w )w0 is the group velocity. The total refractive index n is defined by n2 = n20 + 2n0n2 |A(t)|2, where n2 is the key parameter called the nonlinear refractive index. This coefficient is responsible for a host of nonlinear effects: self- and crossphase modulation, self-focusing, and the optical Kerr effect, to name the important effects. Equation (2) was derived using the following approximations: (1) linearly polarized electric field, (2) homogeneous radial fields, (3) slowly varying envelope, (4) isotropic and nonmagnetic medium, (5) negligible Raman effect, (6) frequency-independent nonlinear susceptibility c(3), and (7) neglect of group velocity dispersion, absorption, self-steepening, and self-frequency shift. Denoting by a and a the amplitude and phase of the electric field envelope A = aeia, Eq. (2) reduces to ∂a 1 ∂a + =0 ∂ z vg ∂ t

(3a)

∂a 1 ∂a w 0 n2 2 + = a . ∂ z vg ∂ t 2c

(3b)

and

The analytical solutions for the amplitude and phase are a(t ) = a0 F (t )

(4a)

and a (z,t ) =

w 0 n2 2c

Ú

z

0

a 2 dz ¢ =

w 0 n2 2 2 a0 F (t )z, 2c

(4b)

where a0 is the amplitude, F(t) the pulse envelope, and t the local time t = t - z/␯g. For materials whose response time is slower than pure electronic but faster than molecular orientation (i.e., coupled electronic, molecular

36

Q.Z. Wang, P.P. Ho, and R.R. Alfano

redistribution, libratory motion) the envelope is just the optical pulse shape. For a “pure” electronic response, the envelope should also include the optical cycles in the pulse shape. The electric field envelope solution of Eq. (2) is given by È w 0 n2 2 2 ( ) ˘ A(z,t ) = a(t ) exp Íi a0 F t z ˙ . Î 2c ˚

(5)

The main physics behind the supercontinuum generation by self-phase modulation is contained in Eq. (5) and is displayed in Figure 2.1. As shown in Figure 2.1a, the index change becomes time dependent and, therefore, the phase of a pulse propagating in a distorted medium becomes time dependent, resulting in self-phase modulation. The electric field frequency is continuously shifted (Figure 2.1c) in time. This process is most important in the generation of femtosecond pulses (see Chapter 10 by Johnson and Shank). Since the pulse duration is much larger than the optical period 2p/w0 (slowly varying approximation), the electric field at each position t within the

Figure 2.1. A simple mechanism for SPM for a non-linear index following the envelope of a symmetrical laser pulse: (a) time-dependent nonlinear index change; (b) time rate of change of index change; (c) time distribution of SPM-shifted frequencies w(t) - w 0.

2. Supercontinuum Generation in Condensed Matter

37

pulse has a specific local and instantaneous frequency at given time that is given by w (t ) = w 0 + dw (t ),

(6a)

where dw (t ) = -

∂a w0 ∂ F 2 (t ) =n2 a02 z . ∂t 2c ∂t

(6b)

The dw(t) is the frequency shift generated at a particular time location t within the pulse shape. This frequency shift is proportional to the derivative of the pulse envelope, which corresponds to the generation of new frequencies resulting in wider spectra. Pulses shorter than the excitation pulse can be produced at given frequencies. It was suggested by Y.R. Shen many years ago that Alfano and Shapiro in 1970 most likely produced femtosecond pulses via supercontinuum generation. Figure 2.1c shows the frequency distribution within the pulse shape. The leading edge, the pulse peak, and the trailing edge are red shifted, nonshifted, and blue shifted, respectively. The spectrum of SPM pulses is obtained by taking the Fourier transform of the complex temporal envelope A(z, t): A(W) =

1 2p

Ú

•

-•

A(z,t ) exp[iWt ]dt ,

(7)

where W = w - w 0. The intensity spectrum is given by S (Wz) =

c 2 A(W, z) . 4p

(8)

In practical cases, the phase of A(z, t) is large compare with p, and the stationary phase method leads to Dw (z) max =

È∂ F 2 w0 n2 a02 Í 2c Î ∂t

t1

∂F2 ∂t

˘ ˙ z. t2 ˚

(9)

The intensity S (W, z) =

2 Ê c ˆ Ê 4pc ˆ Ï 2 ( ) ∂ F F t¢ Ì 2 Ë 4p ¯ Ë w 0 n2 z ¯ Ó ∂t

+2

F (t ¢)F (t ¢¢)

[∂ 2 F 2

+ F 2 (t ¢¢) t¢

∂ 2F 2 ∂t 2

t ¢¢

∂t 2 t ¢ ∂ 2 F 2 ∂t 2 t ¢¢ ]

12

w0 È ˘¸ n2 a02 z(F 2 (t ¢) - F 2 (t ¢¢))˙˝, ¥ cos ÍW(t ¢ - t ¢¢) + 2c Î ˚˛

(10)

where Dw max is the maximum frequency spread, t1 and t2 are the pulse envelope inflection points, and t ¢ and t ≤ are the points of the pulse shape that have the same frequency.

38

Q.Z. Wang, P.P. Ho, and R.R. Alfano

An estimate of the modulation frequency dwM can be made by calculating the maximum number of interference minima and dividing this number into the maximum frequency broadening. A straightforward calculation leads to dw M ª 2p

∂F 2 ∂F 2 /F 2 (t 1 ) ª 2p ∂ t t1 ∂t

/F 2 (t 2 ).

(11)

t2

For a Gaussian laser pulse given by F (t ) = exp[-t 2 2t 02 ],

(12)

the modulation frequency of the SPM spectrum is (Alfano, 1972) dw M =

4p t0

or d vM =

2 . t 0c

(13)

Using this relation, the average modulation period of 13 cm-1 corresponds to an initial pulse duration of 5 ps emitted from mode-locked Nd : glass laser. The maximum frequency extent in this case is (Alfano, 1972) Dw max ª

w 0 n2 a02 z . ct 0

(14)

The maximum frequency shift (Eq. (14)) indicates the following salient points: • The frequency extent is inversely proportional to the pumping pulse duration. The shorter the incoming pulse, the greater the frequency extent. The first white light band supercontinuum pulses were generated using picosecond laser pulses (Alfano and Shapiro, 1970a,b). • The spectral broadening is proportional to n2. The supercontinuum generation can be enhanced by increasing the nonlinear refractive index. This is discussed in detail in Section 6. • The spectral broadening is linearly proportional to amplitude a20. Therefore, multiple-excitation laser beams of different wavelengths mat be used to increase the supercontinuum generation. This leads to the basic principle behind IPM and XPM. These processes are described by Baldeck et al. (Chapter 4) and Manassah (Chapter 5). • The spectral broadening is proportional to w 0 and z. The chirp—the temporal distribution of frequency in the pulse shape— is an important characteristic of SPM broadened pulse. In the linear chirp approximation, the chirp coefficient C is usually defined by the phase relation a = Ct 2 .

(15)

For a Gaussian electric field envelope and linear approximation, the envelope reduces to F 2 (t ) = exp[-t 2 t 02 ] ª 1 - t 2 t 02 . The linear chirp coefficient derived from Eqs. (5) and (16) becomes

(16)

2. Supercontinuum Generation in Condensed Matter 39

Figure 2.2. Calculated SPM spectrum: (a) for response time slower than pure electronic but faster than molecular orientation: b = 30 and t = 0.1 ps; (b) for pure electronic response: b = 30 and t = 0.1 ps. (From Alfano, 1972.)

40

Q.Z. Wang, P.P. Ho, and R.R. Alfano

C=

2 Ê w 0 ˆ Ê n2 a0 z ˆ . Á Ë 2c ¯ Ë t 02 ˜¯

(17)

Typical calculated SPM spectra are displayed in Figure 2.2. The spectral densities of the SPM light are normalized and b is defined as b = (n2a20w 0z)/2c, which measures the strength of the broadening process. Figure 2.2a shows the spectrum for a material response time slower than pure electronic but faster than molecular orientation for b = 30 and t = 0.1 ps. The extent of the spectrum is about 7000 cm-1. Figure 2.2b shows the SPM spectrum for a quasi-pure electronic response for b = 30 and t = 0.1 ps. Typical SPM spectral characteristics are apparent in these spectra.

3. Experimental Arrangement for SPM Generation To produce the supercontinuum, an ultrafast laser pulse is essential with a pulse duration in the picosecond and femtosecond time region. A modelocked laser is used to generate picosecond and femtosecond light pulses. Table 2.2 lists the available mode-locked lasers that can produce picosecond and femtosecond laser pulses. Measurements performed in the 1970s used a modelocked Nd : glass laser with output at 1.06 mm with power of ~5 ¥ 109 W and the second harmonic (SHG) at 530 nm with power of 2 ¥ 108 W. Typically, one needs at least a few microjoulis of 100-fs pulse passing through a 1-mm sample to produce continuum. A typical experimental setup for ultrafast supercontinuum generation is shown in Figure 2.3. Both spectral and spatial distributions are measured. The 8-ps SHG pulse of 5 mJ is reduced in size to a collimated 1.2-mmdiameter beam across the sample by an inverted telescope. For weaker excitation pulses, the beam is focused into the sample using a 10- to 25-cm focal lens. The typical sample length used is 10 to 15 cm for picosecond pulses and 0.1 to 1 cm for 100-f pulses. The intensity distribution of the light at the exit face of the sample was magnified 10 times and imaged on the slit of a specTable 2.2. Available ultrafast mode-locked lasers. Oscillator Ruby YAG Silicate glass Phosphate glass Dye Dye + CO2 + semiconductor switches Dye (CPM) Dye + pulse compression (SYNC) Dye + CPM (prisms in cavity) Dye + SPM + pulse compression (prisms and grating pairs)

Wavelength (nm) 694.3 1064 1060 1054 Tunable (SYNC or flash lamp) 9300 610–630 Tunable 620 620

Pulse duration 30 ps 30 ps 8 ps 6 ps 5–10 ps 1–10 ps 100 fs 300 fs 27–60 fs 6–10 fs

2. Supercontinuum Generation in Condensed Matter

41

Figure 2.3. Experimental arrangement for generating and observing supercontinuum and self-focusing. (From Alfano, 1972.)

trograph. The spectrum of each individual filament within the slit was displayed. Usually there were 5 to 20 filaments. A thin quartz-wedge beam splitter was used to photograph filament formation of the Stokes (anti-Stokes) side of the spectra; three type 3-68 and three type 3-67 (two type 5-60) Corning filters were used to prevent the 530-nm direct laser light from entering the spectrograph. To reduce nonfilament light, a wire 2 mm in diameter was sometimes placed at the focal point of the imaging lens. Previously, spectra were taken on Polaroid type 57 film. At present, video systems such as an Siliconintensified target (SIT) camera together with a PC computer are commonly used to display the spectra. Today, to obtain temporal information about the supercontinuum, a streak camera is added to the experimental system.

4. Generation of Supercontinuum in Solids In the following sections, we review the experimental measurements of supercontinuum generated in condense matter. Topics discussed include supercontinuum generation in various kinds of solids and liquids, optical glass fibers, liquid argon, liquid and solid krypton, magnetic crystals, and dielectric crystals. The mechanisms behind SPM are discussed in Chapter 1 by Shen and Yang. In general, various mechanisms are responsible for SPM in condensed matter and give rise to the coefficient of the intensity-dependent refractive index n2. These are the orientational Kerr effect, electrostriction, molecular redistribution, librations, and electronic distribution. In suitably chosen media (central-symmetric molecules) these frequency-broadening mechanisms may be distinguished from the electronic mechanism through their dif-

42

Q.Z. Wang, P.P. Ho, and R.R. Alfano

ferent time responses (Lallemand, 1966). The relation times for these mechanisms are given approximately by (Brewer and Lee, 1968) t (orientation) = t (molecular types) =

4p 3 ha kT > 10 -12 s, 3

(18)

x2 6pha x 2 = > 10 -12 s, D kT

(19)

t (libration about field) = t (electronic) =

2I >10 -12 s, aE 02

2pa0 h > 1.5 ¥ 10 -16 s, e2

(20) (21)

where h is the viscosity (h = 0.4 cp for liquids and h = 106 cp for glasses); a is the molecular radius; D is the diffusion coefficient (≥10-5 cm/s for liquids) and x is the diffusion distance of the clustering, ~10-8 cm; I is the moment of inertia, Iargon = 9.3 ¥ 10-38 esu and ICCl4 = 1.75 ¥ 10-38 esu; a is the polarizability, aargon = 1.6 ¥ 10-24 esu and aCCl4 = 1.026 ¥ 10-24 esu; and E0 is the amplitude of the electric field, taken as 105 esu, which is close to the atomic field. The response time for an electron distortion is about the period of a Bohr orbit, ~1.5 ¥ 10-16 s. Thus, typical calculated relaxation time responses for diffusional motions are >10-12 s, while the electronic distortion response time is ~150 as. With picosecond light pulses Brewer and Lee (1968) showed that the dominant mechanism for filament formation should be electronic in very viscous liquids. Molecular rocking has been suggested as the cause of broadening and self-focusing in CS2. The molecules are driven by the laser field to rock about the equilibrium position of a potential well that has been set up by the neighboring molecules. This mechanism is characteristized by a relaxation time: t1 =

h = 2.3 ¥ 10 -13 s, G

(22)

where G is the shear modulus ~1.5 ¥ 1010 dynes/cm and viscosity h = 3.7 ¥ 10-3 p for CS2. In solids, mechanisms giving rise to the coefficient of the intensitydependent refractive index n2 for picosecond pulse excitation are either direct distortion of electronic clouds around nuclei or one of several coupled electronic mechanisms: librational distortion, where electronic structure is distorted as the molecule rocks; electron-lattice distortion, where the electron cloud distorts as the lattice vibrates; and molecular distortion, where electronic shells are altered as the nuclei redistribute spatially. The electrostriction mechanism is rejected because it exhibits a negligible effect for picosecond and femtosecond pulses.

2. Supercontinuum Generation in Condensed Matter

43

Typical supercontinuum spectra generated in solids and liquids using 8-ps pulses at 530 nm are displayed in Figure 2.4. All continuum spectra are similar despite the different materials.

4.1 Supercontinuum in Glasses Spectra from the glass samples show modulation (see Figure 2.4a). The spectral modulation ranged from as small as a few wave numbers to hundreds of wave numbers. The filament size was approximately 5 to 50 mm. Typically, 5 to 20 small-scale filaments were observed. Occasionally, some laser output pulses from the samples did not show modulation or had no regular modulation pattern. Typical Stokes sweeps from these filaments were 1100 cm-1 in extradense flint glass of length 7.55 cm and 4200 cm-1 in both borosilicate crown (BK-7) and light barium (LBC-1) glass of length 8.9 cm. Sweeps on the anti-Stokes side were typically 7400 cm-1 in BK and LBC glasses. The sweep is polarized in the direction of the incident laser polarization for unstrained glasses.

4.2 Supercontinuum in Quartz SPM spectra from quartz using an 8-ps pulse at 530 nm are similar to the spectra from glasses displayed in Figure 2.4a. Typical Stokes sweeps from the filaments were 3900 cm-1 in a quartz crystal of length 4.5 cm, and the antiStokes sweeps were 5500 cm-1.

4.3 Supercontinuum in NaCl Sweeps of 3900 cm-1 in NaCl of length 4.7 cm to the red side of 530 nm were observed. Sweeps on the anti-Stokes side were about 7300 cm-1. Some of the spectra show modulation with ranges from a few wave numbers to hundreds of wave numbers. Some laser shots showed no modulation or no regular modulation pattern. For unstrained NaCl, the supercontinuum light is polarized in the direction of the incident laser polarization.

4.4 Supercontinuum in Calcite Sweeps of 4400 cm-1 and 6100 cm-1 to the Stokes and anti-Stokes sides of 530 nm were observed in a calcite crystal of length 4.5 cm (see Figure 2.4b). Some spectra showed modulation structure within the broadened spectra; some showed no modulation or no regular modulation pattern. The exit supercontinuum light has same polarization as the incident laser. The SRS threshold is lower for laser light traveling as an O-wave than an E-wave. SPM dominates the E-wave spectra.

44

Q.Z. Wang, P.P. Ho, and R.R. Alfano

Figure 2.4. Supercontinuum spectra from various kind of solids and liquids. (a) Stokes and anti-Stokes SPM from BK-7 glass and filament formation for different laser shots. The filaments are viewed through Corning 3-67 filters. (b) Stokes and anti-Stokes SPM from calcite for different laser shots. The laser beam propagates as an O-wave through the sample. (c) Stokes and anti-Stokes SPM spectra from calcite for different laser shots. The laser beam propagates as an E-wave. (From Alfano, 1972.)

2. Supercontinuum Generation in Condensed Matter

Figure 2.4. (continued )

45

46

Q.Z. Wang, P.P. Ho, and R.R. Alfano

Figure 2.4. (continued )

2. Supercontinuum Generation in Condensed Matter

47

Figure 2.5. Relative emission intensity versus emission wavelength for KBr. Exciting wavelength = 1.06 mm. (From Yu et al., 1975.)

4.5 Supercontinuum in KBr A high-power broadband coherent source in the near- and medium-infrared region can be realized by passing an intense 1.06-mm picosecond pulse through a KBr crystal. Figure 2.5 shows the spectra from 10-cm-long KBr crystal with excitation of a 9-ps, 1011 W/cm2 pulse at 1.06 mm. On the Stokes side the maximum intensity occurs at 1.2 mm. When the signal drops to 10-1 the span of the spectral broadening is Dvs = 3200 cm-1 on the Stokes side and Dva = 4900 cm-1 on the anti-Stokes side. Beyond 1.6 mm the signal level falls off rapidly. At 1.8 mm the signal is 10-2 and at 2 mm no detectable signal can be observed (Yu et al., 1975).

4.6 Supercontinuum in Semiconductors Infrared supercontinuum spanning the range 3 to 14 mm can be obtained when an intense picoseond pulse generated from a CO2 laser is passed into GaAs, AgBr, ZnSe, and CdS crystals (Corkum et al., 1985). The supercontinuum spectra measured from a 6-cm-long Cr-doped GaAs crystal and a 3.8-cm AgBr crystal for different laser pulse durations and

48

Q.Z. Wang, P.P. Ho, and R.R. Alfano

Figure 2.6. Supercontinuum spectra from a 6-cm-long Cr-doped GaAs crystal. (From Corkum et al., 1985.)

intensities and plotted in Figures 2.6 and 2.7, respectively. The signals were normalized for the input laser energy and the spectral sensitivity of filters, grating, and detectors. Each point represents the average of three shots. The salient feature of the curves displayed in Figures 2.6 and 2.7 is that the spectral broadening spans the wavelength region from 3 to 14 mm. The wave number spread on the anti-Stokes side is much greater than that on the Stokes side. From data displayed in Figure 2.6, the maximum anti-Stokes spectral broadening is Dwa = 793 cm-1. Including second and third harmonic generation (SHG and THG), it spans 2000 cm-1. On the Stokes side, Dws = 360 cm-1, yielding a value of dwa/dws ~ 2.2. For AgBr, Figure 2.7 shows that Dwa = 743 cm-1 and Dws = 242 cm-1, yielding Dwa/Dws ~ 3. The spectral broadening mechanism for the supercontinuum can originate from several nonlinear optical processes. These include self-phase modulation, the four-wave parametric effect, higher-order harmonic generation, and stimulated Raman scattering. In Figure 2.6 the supercontinuum from the GaAs has two small peaks at 4.5 and 3.3 mm. These arise from the SHG and THG, respectively. Small plateaus are located at 7.5 and 12 mm. These arise from the first-order anti-Stokes and Stokes stimulated Raman scattering combined with SPM about these wavelengths. The SPM is attributed to an electronic mechanism.

2. Supercontinuum Generation in Condensed Matter

49

Figure 2.7. Supercontinuum spectra from a 3.8-cm-long AgBr crystal. (From Corkum et al., 1985.)

Summarizing the important experimental aspects of the spectra in condensed matter: The spectra are characterized by very large spectral widths and a nonperiodic or random substructure. Occasionally, a periodic structure interference minimum and maximum are observed. The modulation frequencies range from a few cm-1 to hundreds of cm-1, and some modulation progressively increases away from the central frequency. The Stokes and antiStokes spectra are approximately equal in intensity and roughly uniform. The extents on the Stokes and anti-Stokes sides are not symmetric. The peak intensity at the central frequency is 102–103 the intensity of the SPM spectra at a given frequency.

5. Generation of Supercontinuum in Liquids Nonlinear optical effects in solids are very effective; however, damage generated in solid media often limits their usefulness for ultrashort high-power effects. Various kinds of inorganic and organic liquids are useful media for generating picosecond or femtosecond supercontinuum light pulses since they are selfhealing media. The supercontinuum spectra produced in liquids (Alfano, 1972) are similar to the spectra displayed in Figure 2.4 (Alfano,

50

Q.Z. Wang, P.P. Ho, and R.R. Alfano

1972). The following highlights the supercontinuum phenomena in the various favorite liquid media of the authors. These liquids give the most intense and uniform supercontinuum spectral distributions.

5.1 Supercontinuum in H2O and D2O The supercontinuum generated in H2O and D2O by the second harmonic of a mode-locked neodymium glass laser spanned several thousand wave numbers. The time duration was equal to or less than the picosecond pulse that generated it (Busch et al., 1973). The continuum extended to below 310 nm on the anti-Stokes side and to the near-IR region on the Stokes side. There were sharp absorptions at 450 nm in the H2O continuum and at 470 nm in the D2O continuum resulting from the inverse Raman effect (Alfano and Shapiro, 1970b; Jones and Stoicheff, 1964). Focusing a 12-mJ, 1060-nm single pulse 14 ps in duration into 25 cm of liquid D2O resulted in a continuum that showed practically no structure, extending from 380 to at least 800 nm and highly directional and polarized (Sharma et al., 1976). Enhancing the supercontinuum intensity using water with ions is discussed in Section 10.

5.2 Supercontinuum in CCl4 Another favorite liquid for producing a supercontinuum is CCl4, in which the spectra produced are similar to the spectra displayed in Figure 2.4. A typical flat white supercontinuum extending from 430 nm through the visible and near infrared could be produced by focusing an 8-ps pulse at 1060 nm with about 15 mJ pulse energy into a cell containing CCl4 (Magde and Windsor, 1974).

5.3 Supercontinuum in Phosphoric Acid Orthophosphoric acid was found to be a useful medium for generating picosecond continuum light pulses ranging from the near UV to the near IR. By focusing a pulse train from a mode-locked ruby laser into a 10-cm-long cell containing phosphoric acid (60% by weight) solution in water by an 8-cm focal lens, a supercontinuum from near 450 nm to the near IR was obtained. The supercontinuum spectra contain structure arising from Raman lines (Kobayashi, 1979).

5.4 Supercontinuum in Polyphosphoric Acid The supercontinuum from polyphosphoric acid was generated by focusing an optical pulse at 694.3 nm with 100 mJ pulse energy and a pulse width of 28 ps into a cell of any length from 2 to 20 cm containing polyphosphoric acid. It reaches 350 nm on the anti-Stokes side, being limited by the absorp-

2. Supercontinuum Generation in Condensed Matter

51

tion of polyphosphoric acid, and 925 nm on the Stokes side, being cut off by limitations of IR film sensitivity (Nakashima and Mataga, 1975).

6. Supercontinuum Generated in Optical Fibers The peak power and the interaction length can be controlled better in optical fibers than in bulk materials. Optical fibers are particularly interesting material for nonlinear optical experiments. In this section, we discuss supercontinuum generation in glass optical fibers. Details of the use of SPM for pulse compression are discussed in other chapters. The generation of continua in glass optical fibers was performed by Stolen et al. in 1974. Continua covering ~500 cm-1 were obtained. Shank et al. (1982) compressed 90-fs optical pulses to 30-fs pulses using SPM in an optical fiber followed by a grating compressor. Using the SPM in an optical fiber with a combination of prisms and diffraction gratings, they were able to compress 30 fs to 6 fs (Fork et al., 1987; also see Chapter 10 by Johnson and Shank). A typical sequence of spectral broadening versus input peak power using 500-fs pulses (Baldeck et al., 1987b) is shown in Figure 2.8. The spectra show

Figure 2.8. Sequence of spectral broadening versus increasing input energy in a single-mode optical fiber (length = 30 cm). The intensity of the 500-fs pulse was increased from (a) to (c). (From Baldeck et al., 1987b.)

52

Q.Z. Wang, P.P. Ho, and R.R. Alfano

Figure 2.9. Supercontinuum spectra versus input pulse energy in a single-mode optical fiber (length = 30 cm) for a 500-fs pulse. (From Baldeck et al., 1987b.)

SPM characteristic of heavy modulation. The spectral extent is plotted against the energy in Figure 2.9 for 500-fs pulses (Baldeck et al., 1987b). The relative energy of each pulse was calculated by integrating its total broadened spectral distribution. The supercontinuum extent increased linearly with the pulse intensity. The fiber length dependence of the spectral broadening is plotted in Figure 2.10. The broadening was found to be independent of the length of the optical fiber for l > 20 cm. This is due to group velocity dispersion. The SPM spectral broadening occurs in the first few centimeters of the fiber for such short pulses (Baldeck et al., 1987b). In multimode optical fibers, the mode dispersion is dominant and causes pulse distortion. Neglecting the detailed transverse distribution of each mode, the light field can be expressed by E (t) = Â ai Ai (t) exp[iw 0t - iki z],

(23)

i

where w0 is the incident laser frequency; ai, Ai(t), and ki = niw0/c are the effective amplitude, electric field envelope function at the local time t = t - z/vgi, and wave number of mode i, respectively; and vgi is the group velocity of mode i. The effective refractive index of mode i is denoted by ni and ni = n0i + n2i E (t) , 2

(24)

where n0i and n2i are the linear refractive index and the nonlinear coefficient of the ith mode, respectively. The nonlinearities of different modes are

2. Supercontinuum Generation in Condensed Matter

53

Figure 2.10. Supercontinuum spectra versus optical fiber length for a 500-fs pulse. (From Baldeck et al., 1987b.)

assumed to be the same, that is, n2i = n2. Substituting Eq. (21) into Eq. (20), we obtain E (t) = Â ai Ai (t) exp[iw 0t - in0i w 0 z c - iDf (t)],

(25)

Df (t) = (n2w 0 z c) E (t) .

(26)

i

where 2

After inserting Eq. (23) into Eq. (26), the time-dependent phase factor Df(t) can be expanded in terms of Ei(t): Df (t) = Â Â (n2w 0 z c)ai a j Ai (t)A j (t) exp[i ( n0i - n0 j ) w 0 z c]. i

(27)

j

In the picosecond time envelope, the terms of i π j oscillate rapidly. Their contributions to the time-dependent phase factor are washed out. The approximate Df(t) has the form Df = Â (n2w 0 z c)ai2 Ai2 (t).

(28)

i

The pulse shape changes due to the different group velocities of various modes. When most of the incident energy is coupled into the lower modes, the pulse will have a fast rising edge and a slow decay tail since the group velocity is faster for lower-order mode. This feature was observed using a streak camera. Therefore, the Df(t, z) of Eq. (28) will also have a fast rising edge and a slow decay tail. The time derivative of the phase Df(t, z) yields an asymmetric frequency broadening. Figure 2.11 shows the spectral continuum generated from multimode glass optical fibers using 8-ps pulses at 530 nm. The spectral broadening is asym-

54

Q.Z. Wang, P.P. Ho, and R.R. Alfano Figure 2.11. Output spectra for 8-ps laser pulses at 527 nm propagating through different lengths of multimode optical fibers: (a) no fiber; (b) 22 cm; (c) 42 cm; (d) 84 cm. (From Wang et al., 1988.)

metric about the incident laser frequency. It is shifted much more to the Stokes side than to the anti-Stokes side. The observed spectra did not show a modulation. This can be explained by the spectral resolution of the measurement system. The calculated modulation period is about 0.13 nm, which is much smaller than the resolution of the measurement system (about 1 nm) (Wang et al., 1988).

7. Supercontinuum Generation in Rare-Gas Liquids and Solids Continuum generation is a general phenomenon that occurs in all states of matter. A system for testing the role of the electronic mechanism is rare-gas liquids and solids (Alfano and Shapiro, 1970a). Rare-gas liquids are composed of atoms possessing spherical symmetry. Thus, there are no orientational, librational, or electron-lattice contributions to the nonlinear refractive index n2. However, interrupted rocking of argon can occur in which a distorted atom can rock about an equilibrium value before it collides with other atoms. Contributions to the nonlinear refractive index might be expected from electrostriction, molecular redistribution, interrupted rocking, and a distortion of the electron clouds: n2 = n2 ELECTRONIC + n2 MR + n2 LIBRATION + n2 ELECTROSTR .

(29)

Electrostriction is ruled out because picosecond exciting pulses are too short. Molecular redistribution arises from fluctuations in the local positional

2. Supercontinuum Generation in Condensed Matter

55

arrangement of molecules and can contribute significantly to n2. However, n2 due to all mechanisms except electronic was estimated to be ~2 ¥ 10-14 esu for liquid argon from depolarized inelastic-scattering data. Electronic distortion (n2 = 6 ¥ 10-14 esu) slightly dominates all nonlinear index contributions (Alfano and Shapiro, 1970a; Alfano, 1972). Furthermore, the depolarized inelastic light-scattering wing vanishes in solid xenon, implying that the molecular redistribution contribution to n2 vanishes in rare-gas solids. Observations of self-focusing and SPM in rare-gas solids appear to provide a direct

Figure 2.12. Supercontinuum spectra for picosecond laser pulses at 530 nm passing through rare-gas liquids and solids; (a) Stokes SPM from liquid argon for different laser shots; (b) anti-Stokes SPM for liquid argon for different laser shots; (c) Stokes SPM for liquid and solid krypton for different laser shots. (From Alfano, 1972.)

56

Q.Z. Wang, P.P. Ho, and R.R. Alfano

Figure 2.12. (continued )

proof that atomic electronic shells are distorted from their spherical symmetry under the action of the applied field. However, both pure electronic and molecular redistribution mechanisms contribute to n2 in rare-gas liquids. The response time of the system for a combination of both of these mechanisms lies between 10-15 and 10-12 s. For femtosecond and subpicosecond pulses, the dominant mechanism for n2 and SPM is electronic in origin. The experimental setup used to generate and detect a supercontinuum in rare-gas liquids and solids is the same as that shown in Figure 2.3 with the exception that the samples are placed in an optical dewar. Typical supercontinuum spectra from rare-gas liquids and solids are displayed in Figure 2.12. Sweeps of 1000 to 6000 cm-1 were observed to both the Stokes and anti-Stokes sides of 530 nm in liquid argon. Modulation ranges from a few cm-1 to hundreds of cm-1. Similar spectral sweeps were observed in liquid and solid krypton.

2. Supercontinuum Generation in Condensed Matter

57

Figure 2.12. (continued)

A most important point is that the threshold for observing SPM in liquid krypton is 0.64 ± 0.12 that in liquid argon. The SPM threshold ratio of solid and liquid krypton is 0.86 ± 0.15. In liquid argon, SPM spectra appear at a threshold power of ~0.5 GW focused in a 12-cm sample. The swept light is also collimated, polarized, and modulated. These observations rule out dielectric breakdown. The refractive index in rare-gas liquids is given by n|| = n0 + n2·E2Ò, where n|| is the refractive index parallel to the field. ·E2Ò1/2 is the rms value of the electric field. The electronic nonlinear refractive index in rare gas liquids is given by

[

]

n2 = (n02 + 2) 81n0 pNr, 4

(30)

where n0 is the linear refractive index, r is the second-order hyperpolarizability, and N is the number of atoms per unit volume. The term n2 = 0.6 ¥ 10-13 esu in liquid argon and ⯝1.36 ¥ 10-13 in liquid krypton. For liquid argon

58

Q.Z. Wang, P.P. Ho, and R.R. Alfano

and liquid and solid krypton, the refractive indices are taken as 1.23, 1.30, and 1.35, respectively (McTague et al., 1969). Intense electric fields distort atoms and produce a birefringence. The anisotropy in refractive index between light traveling with the wave vector parallel and perpendicular to the applied electric field is given by (Alfano, 1972) dn|| - dn^ = 31 n2 E 02 ,

(31)

where dn|| and dn⬜ are the changes in refractive indices parallel and perpendicular to the field. The value of n2E20 is ~5 ¥ 10-5 V/m in liquid argon when E0 ~ 1.5 ¥ 107 V/m (~4 ¥ 1011 W/cm2). This change in index explains the self-focusing and SPM described above which was observed by Alfano and Shapiro in 1970. Similar SPM effects occur in organic and inorganic liquids, often accompanied by SRS and inverse Raman effects.

8. Supercontinuum Generation in Antiferromagnetic KNiF3 Crystals The influence of magnetic processes on nonlinear optical effects is an interesting topic. In this section, we discuss the supercontinuum generation associated with the onset of magnetic order in a KNiF3 crystal (Alfano et al., 1976). Light at 530 nm is well suited for the excitation pulse because KNiF3 exhibits a broad minimum in its absorption (Knox et al., 1963) between 480 and 610 nm. Typical spectra from an unoriented 5-cm-long KNiF3 single crystal are displayed in Figure 2.13 for 530-nm picosecond excitation (Alfano et al., 1976). The spectra are characterized by extensive spectral broadening ranging up to ~3000 cm-1 to either side of the laser frequency. The intensity, although not the spectral broadening, of the output exhibited the large temperature dependence illustrated in Figure 2.14. There is no sharp feature at 552 nm, the position expected for stimulated Raman scattering by the 746-cm-1 magnon pair excitation. Usually, the spectra were smooth; however, occasionally structure was observed. A periodic structure with a modulation frequency of tens to hundreds of wave numbers was evident. The frequency broadening light is also polarized in the same direction as the incident 530-nm pulse. This property is the same observed in glass, crystals, and liquids (see Sections 3–6). Self-focusing was also observed, usually in the form of 10 to 40 small selffocused spots 5 to 20 mm in diameter at the exit face of the crystal. Using a focused beam, optical damage could also be produced. It should be emphasized that spectral broadening was always observed even in the absence of self-focusing, damage, or periodic spectral intensity modulation. Figure 2.15 shows the output intensity at 570 nm as a function of input intensity for two temperatures: above and below the Néel temperature. The output intensity is approximately exponential in the input intensity at both

2. Supercontinuum Generation in Condensed Matter

59

Figure 2.13. Spectra for picosecond laser pulse at 530 nm passing through 5-cm-long KNiF3. (From Alfano et al., 1976.)

Figure 2.14. Intensity of the frequency-broadening emission from KNiF3 as a function of temperature at fixed pump intensity at 552 nm. (From Alfano et al., 1976.)

60

Q.Z. Wang, P.P. Ho, and R.R. Alfano

Figure 2.15. Intensity dependence of continuum spectra at 570 nm from KNiF3 as a function of pumping laser intensity at fixed lattice temperature. (From Alfano et al., 1976.)

temperatures. However, the slope is more than a factor of two larger at 77 K than at 300 K. The rapid rise in conversion efficiency of four orders of magnitude within a small interval of input intensity is indicative of an amplification process with very large gain. Identical curves were obtained at 552 and 600 nm output wavelengths. The similarity in results for several output frequencies shows that simple stimulated magnon pair scattering is not the

2. Supercontinuum Generation in Condensed Matter

61

dominant process. If it were, one would expect the behavior at 552 nm to differ considerably from that at other wavelengths. The most novel experimental results in KNiF3 are the large (~20¥) intensity increases below TN. Spectra at 552, 570, and 600 nm behave identically— within experimental error—consistent with the observations in Figure 2.15. The temperature dependence of the relative peak intensity for the spontaneous magnon pair scattering in the KNiF3 sample (using 514.5 nm laser light) was measured and is potted in Figure 2.14. For KNiF3 the magnon pair scattering accounts for the entire inelastic light scattering and therefore for the non-s electronic contribution to c3 (Hellwarth et al., 1975). The temperature dependence is compelling evidence for the magnetic origin of the lowtemperature-enhanced nonlinear optical spectral broadened intensity. The observation can be semiquantitatively accounted for in terms of a temperature-dependent spin contribution to the overall nonlinear susceptibility c(3) ijkl that governs four-photon parametric mixing as the primary process. In general c(3) may be written as a sum of electronic and Raman contributions (Levenson and Bloembergen, 1974). For KNiF3 we may consider the latter to consist solely of the magnon pair Raman scattering contribution (Chinn et al., 1971; Fleury et al., 1975), which we can approximate as a Lorentzian: (3 ) (-w 3 ,w 1,w 1, -w 2 ) = c E(3) + K c ijkl

a ijma klm + a ilma mjk . w m - (w 1 - w 2 ) + iGm

(32)

Here wm and Gm denote the temperature-dependent frequency and linewidth, respectively, of the magnon pair excitations, a ijm is the magnon pair polarizability, and cE(3) is the usual nonresonant, temperature-independent “electronic” contribution from nonlinear distortion of the electronic orbits. The second term in Eq. (32) is called magnetic cM(3). Since the integrated intensity of the spontaneous magnon pair Raman spectrum, which is ~|a m|2, has been measured and found to be essentially temperature independent (Chinn et al., 1971; Fleury et al., 1975), the only quantities in Eq. (32) that vary significantly with temperature are wm and Gm. The observed temperature independence of the extent of spectral broadening, dw, may be explained by noting that dw ~ 2Dwn2kE 21l due to self-phase modulation. Here Dw is the spectral width of the input pulse, k is its propagation constant, E1 is the field amplitude, and l is the path length. n2 is the nonlinear refractive index, which contains a purely electronic contribution, s, and a contribution proportional to the integrated Raman scattering cross section (Hellwarth et al., 1975). Since neither s nor |a m|2 is temperature dependent in KNiF3, n2 and therefore Dw should not vary either, in agreement with observations. The observed strong temperature dependence of the intensity of the frequency-broadened spectrum (see Figure 2.14) arises from the resonant term in Eq. (32) through the primary process 2w1 Æ w 2 + w 3, which is strongest when w 2 = w 1 + w m and increases as Gm decreases (on cooling below the Néel

62

Q.Z. Wang, P.P. Ho, and R.R. Alfano

temperature). That is, the resonant contribution to c(3) in Eq. (32) varies with temperature in the same way as the peak spontaneous magnon pair cross section: Gm-1(T ). However, the individual contribution to c(3) cannot be directly inferred from the dependence of the broadened spectrum. This is because the latter receives significant contributions from secondary processes of the form w1 + w¢2 Æ w 3¢ + w4 etc., in which products of the primary process interact with the pump to smooth the spectral distribution and wash out the sharp features that the resonant spin nonlinearity produces in the primary process. The large values of pump intensity and source spectral width make possible strong amplification in spite of imprecise phase matching in the forward direction. Such behavior (washing out of stimulated Raman features by the spectral broadening process) has frequently been observed in both liquids and crystals. Thus a full quantitative description of the nonlinear optical processes in KNiF3 is not yet possible.

9. Generation of Supercontinuum near Electronic Resonances in Crystals Since the active medium of a laser possesses well-defined electronic energy levels, knowledge of SPM near electronic levels is of paramount importance. SPM near electronic levels of a PrF3 crystal has been investigated experimentally and theoretically to gain additional information on the SPM process—in particular, on the role played by the electronic levels and on how the continuum spectrum evolves through and beyond the electronic absorption levels (Alfano et al., 1974). Experimentally, the Stokes and anti-Stokes spectrum and filament formation from the PrF3 crystal are investigated under intense picosecond pulse excitation at the wavelength of 530 nm. The c axis of the crystal is oriented along the optical axis. The intensity distribution at the exit face of the crystal is magnified by 10¥ and imaged on the slit of a Jarrell-Ash –12 -m-grating spectrograph so that the spectrum of each filament can be displayed. The spectra are recorded on Polaroid type 57 film. No visible damage occurred in the PrF3 crystal. The PrF3 crystal was chosen for the experiment because its electronic levels are suitably located on the Stokes and anti-Stokes sides of the 530-nm excitation wavelength. The absorption spectra of a –12 -mm-thick PrF3 crystal and the energy level scheme of Pr3+ ions are shown in Figure 2.16. The fluorides of Pr have the structure of the naturally occurring mineral tysonite with D434 symmetry. Typical spectra of frequency broadening from PrF3 about 530 nm are shown in Figure 2.17 for different laser shots. Because of the absorption associated with the electronic level, it is necessary to display the spectrum over different wavelength ranges at different intensity levels. In this manner, the

2. Supercontinuum Generation in Condensed Matter

63

Figure 2.16. Absorption spectra of 0.5-mm-thick PrF3 crystal; insert is the level scheme of Pr3+ ions. (From Alfano et al., 1974.)

development of the SPM spectrum through the electronic absorption levels can be investigated. Using appropriate filters, different spectral ranges are studied and displayed in the following figures: in Figure 2.18a the Stokes side for frequency broadening v–B > 100 cm-1 at an intensity level (ISPM) of ~10-2 of the laser intensity (IL), in Figure 2.18b the Stokes side for v–B > 1500 cm-1 at ISPM ~ 10-4 IL, in Figure 2.18c the anti-Stokes side for v–B > 100 cm-1 at ISPM ~ 10-2 IL, and in Figure 2.18d the anti-Stokes side for v–B > 1500 cm-1 at ISPM ~ 10-4 IL. Usually 50 to 100 small-scale filaments 5 to 50 mm in diameter are observed. Several salient features are evident in the spectra displayed in Figures 2.17 and 2.18. In Figure 2.17 the Stokes and anti-Stokes spectra are approximately equal in intensity and frequency extent. The peak intensity at the central frequency is ~100 times the intensity of the SPM at a given frequency. The extent of the frequency broadening is ~1500 cm-1, ending approximately at the absorption lines. Occasionally a periodic structure of minima and maxima is observed that ranges from a few cm-1 to 100 cm-1, and for some observations no modulation is observed. Occasionally an absorption band appears on the

64

Q.Z. Wang, P.P. Ho, and R.R. Alfano

Figure 2.17. Spectra from PrF3 excited by 4-ps laser pulses at 530 nm; neutral density (ND) filters: (a) ND = 1.5; (b) ND = 1.5; (c) ND = 2.0; (d) ND = 2.0; (e) ND = 1.7; (f ) ND = 1.4. A wire is positioned after the collection lens at the focal length. (From Alfano et al., 1974.)

Figure 2.18. Spectra on the Stokes and anti-Stokes sides of the 530-nm excitation: (a) Stokes side, Corning 3-68 filter, wire inserted, ND = 2.0; (b) Stokes side, Corning 3-66 filter, wire inserted; (c) anti-Stokes side, wire inserted, ND = 1.0; (d) anti-Stokes side, Corning 5-61, wire inserted. (From Alfano et al., 1974.)

2. Supercontinuum Generation in Condensed Matter

65

Figure 2.19. Comparison of the Stokes absorption spectra of PrF3 photographed with different light sources: (a) light emitted from a tungsten lamp is passed through 0.5-cm-thick crystal; (b) SPM light emitted from BK-7 glass is passed through 0.5-mm-thick crystal; (c) SPM light is generated within the 5-cm PrF3. (From Alfano et al., 1974.)

anti-Stokes side of the 530-nm line whose displacement is 430 cm-1. In Figure 2.18 the main feature is the presence of a much weaker super-broadband continuum whose frequency extends through and past the well-defined absorption lines of the Pr3+ ion to a maximum frequency of >3000 cm-1 on the Stokes side (end of film sensitivity) and >6000 cm-1 on the anti-Stokes side. The intensity of the continuum at a given frequency outside absorption lines is ~10-4 the laser intensity. The observed absorption lines on the anti-Stokes side of 530 nm are located at 441.5, 465.3, and 484.5 nm and on the Stokes side at 593 and 610.9 nm. These lines correspond within ±0.7 nm to the absorption lines measured with a Cary 14. The absorption lines measured from the Cary spectra are ~3 cm-1 at 611.2 nm, 62 cm-1 at 5938.8 nm, 46 cm-1 at 485.2 nm, and >100 cm-1 at 441.2 nm. Figure 2.19 compares the Stokes absorption spectra of a PrF3 crystal photographed with a –12 -m Jarrell-Ash spectrograph with different broadband light sources. Figure 2.19a was obtained with light emitted from a tungsten lamp passing through a –12 -mm PrF3 crystal, Figure 2.19b was obtained with the Stokes side of the broadband picosecond continuum generated in BK-7 glass passing through a –12 -mm PrF3, and Figure 2.19c was obtained with the broadband light generated in a 5-cm PrF3 crystal. Notice that the absorption line at 611.2 nm is very pronounced in the spectra obtained with the continuum generated in PrF3, whereas with conventional absorption techniques it is barely visible. The anti-Stokes spectrum obtained 1 with light emitted from a tungsten filament lamp passing through a –2 -mm PrF3 crystal is shown in Figure 2.20a. This is compared with the spectrum obtained with broadband light generated in a 5-cm PrF3 crystal shown in Figure 2.20b.

66

Q.Z. Wang, P.P. Ho, and R.R. Alfano Figure 2.20. Comparison of the antiStokes absorption spectrum of PrF3 photographed with (a) light emitted from a tungsten lamp passing through 0.5-mmthick crystal and (b) SPM light generated within the 5-cm PrF3. (From Alfano et al., 1974.)

Figure 2.21. Angular variation of the (a) Stokes and (b) anti-Stokes spectral patterns emitted from PrF3 crystal: (a) Corning 4(3-67) filters, ND = 1.0; (b) Corning 2(5-60) filters. (From Alfano et al., 1974.)

The angular variation of the anti-Stokes and Stokes spectral emission from PrF3 is displayed in Figure 2.21. The light emitted from the sample is focused 1 on the slit of a –2 -m Jarrell-Ash spectrograph with a 5-cm focal length lens with the laser beam positioned near the bottom of the slit so that only the upper half of the angular spectrum curve is displayed. In this fashion, a larger angular variation of the spectrum is displayed. Emission angles >9° go off slit and are not displayed. This spectrum is similar to four-photon emission patterns observed from glass and liquids under picosecond excitation. The experimental results show that a discontinuity in intensity occurs when the self-phase modulation frequency extends beyond the absorption line frequency. This is due to almost total suppression of the signal beyond the absorption resonance (Alfano et al., 1974). A similar argument and conclusion hold for the blue side of the laser line. The residual weak intensity that exists beyond the absorption line is not due to SPM. It can arise, however, from three-wave mixing. Since there was a continuum of frequencies created by SPM, it might be possible for three such frequencies, w1, w2, and w3, to mix to create a signal at frequency w1 + w2 - w3 that lies beyond the absorption line. Since the frequencies are chosen from a continuum, it is also possible for phase matching to be achieved. For the spectrum in the domain between the laser frequency and the absorption line, the extent of selfbroadening is proportional to the intensity. Since the energy in the pulse is proportional to the product of the frequency extent and the intensity spectrum, the intensity spectrum remains approximately constant. The observed

2. Supercontinuum Generation in Condensed Matter

67

absorption band in the continuum on the anti-Stokes side about 400 cm-1 away from the excitation frequency (see Figure 2.17) is probably due to the inverse Raman effect (Jones and Stoicheff, 1964). The observed absorption band is located in the vicinity of strong Raman bands: 401, 370, and 321 cm-1. A curious feature of the associated weak broadband spectrum is the existence of a pronounced absorption line at a position (611.2 nm) where the linear absorption would be expected to be rather weak. A possible explanation for this is as follows: Imagine tracing the spatial development of the phase modulation spectrum. At a short distance, where the bounds of the spectrum have not yet intersected a strong absorption line, the spectrum is reasonably flat. On intersecting the absorption line, the spectrum abruptly drops (Alfano et al., 1974). The mechanism of FFPG is presumably responsible for the appearance of the signal beyond the absorption line limit. This explanation is also supported by the appearance of the angular emission pattern (see Figure 2.21). As the spectrum continues to develop, one reaches a point where the limit of the regenerated spectrum crosses a weak absorption line. One can again expect a drastic drop in the spectrum at the position of this line. At still greater distances renewed four-photon parametric regeneration accounts for the feeble signal. A continuum is generated behind absorption bands due to contributions from SPM, three-wave mixing (TWM), and FFPG.

10. Enhancement of Supercontinuum in Water by Addition of Ions The most common liquids used to generate a continuum for various applications are CCl4, H2O, and D2O. In most applications of the ultrafast supercontinuum, it is necessary to increase the conversion efficiency of laser excitation energy to the supercontinuum. One method for accomplishing this is based on the induced- or cross-phase modulation. Another way is to increase n2 in materials. In this section, chemical means are used to obtain a tenfold enhancement of the ultrafast supercontinuum in water by adding Zn2+ or K+ ions (Jimbo et al., 1987) for 8-ps pulse generation. The optical Kerr gate (OKG) (Ho and Alfano, 1979) was used to measure the nonlinear refractive index of the salt solutions. The primary and second harmonic light beams were separated by a dichroic mirror and then focused into a 1-cm-long sample cell filled with the same salt solutions that produced the ultrafast supercontinuum pulse enhancements. The size of the nonlinear index of refraction, n2, was determined from the transmission of the probe beam through the OKG. Three different two-component salt solutions of various concentrations were tasted. The solutes were KCl, ZnCl2, and K2ZnCl4. All measurements were performed at 20 ± 1°C. Typical spectra of ultrafast supercontinuum

68

Q.Z. Wang, P.P. Ho, and R.R. Alfano

pulses exhibited both SPM and FPPG features. The collinear profile arising from SPM has nearly the same spatial distribution as the incident 8-ps, 530-nm laser pulse. The two wings correspond to FPPG pulse propagation. The angle arises from the phase-matching condition of the generated wavelength emitted at different angles from the incident laser beam direction. FPPG spectra sometimes appear as multiple cones and sometimes show modulated features. SPM spectra also show modulated patterns. These features can be explained by multiple filaments. Typical ultrafast supercontinuum pulse spectra on the Stokes side for different aqueous solutions and neat water, measured with the optical multichannel analyzer, are shown in Figure 2.22. The salient features in Figure 2.22 are a wideband SPM spectrum together with the stimulated Raman scattering of the OH stretching vibration around 645 nm. The addition of salts causes the SRS signal to shift toward the longer-wavelength region and sometimes causes the SRS to be weak (Figure 2.22a). The SRS signal of pure water and dilute solution appears in the hydrogen-bonded OH stretching region (~3400 cm-1). In a high-concentration solution, it appears in the nonhydrogen-bonded OH stretching region (~3600 cm-1). The latter features of SRS were observed in an aqueous solution of NaClO4 by Walrafen (1972). To evaluate quantitatively the effect of cations on ultrafast supercontinuum generation, the ultrafast supercontinuum signal intensity for various samples at a fixed wavelength were measured and compared. Figure 2.23 shows the dependence of the supercontinuum (mainly from the SPM contribution) signal intensity on salt concentration for aqueous solutions of K2ZnCl4, ZnCl2, and KCl at 570 nm (Figure 2.23a) and 500 nm (Figure 2.23b). The data were normalized with respect to the average ultrafast supercontinuum signal intensity obtained from neat water. These data indicated that the supercontinuum pulse intensity was highly dependent on salt concentration and that both the Stokes and the anti-Stokes sides of the supercontinuum signals from a saturated K2ZnCl4 solution were about 10 times larger than from neat water. The insets in Figure 2.23 are the same data plotted as a function of K+ ion concentration for KCl and K2ZnCl4 aqueous solutions. Solutions of KCl and K2ZnCl4 generate almost the same amount of supercontinuum if the K+ cation concentration is same, even though they contain different amounts of Cl- anions. This indicates that the Cl- anion has little effect on generation of the supercontinuum. The Zn2+ cations also enhanced the supercontinuum, though to a lesser extent than the K+ cations. The measurements of the optical Kerr effect and the ultrafast supercontinuum in salt-saturated aqueous solutions are summarized in Table 2.3. The measured n2 (pure H2O) is about 220 times smaller than n2 (CS2). The value GSPM(l) represents the ratio of the SPM signal intensity from a particular salt solution to that from neat water at wavelength l. GKerr is defined as the ratio of the transmitted intensity caused by a polarization change of the probe beam in a particular salt solution to that in neat water; GKerr is equal to [n2 (particular solution/n2 (water)]2. Table 2.3 shows that, at saturation, K2ZnCl4

2. Supercontinuum Generation in Condensed Matter

69

Figure 2.22. SPM spectrum of (a) saturated K2ZnCl4 solution, (b) 0.6-m K2ZnCl4, and (c) pure water. The SRS signal (645 nm) is stronger in pure water, and it disappears in high-concentration solution. (From Jimbo et al., 1987.)

70

Q.Z. Wang, P.P. Ho, and R.R. Alfano

Figure 2.23. Salt concentration dependence of the SPM signal (a) on the Stokes side and (b) on the anti-Stokes side 20°C. Each data point is the average of about 10 laser shots. The inserts are the same data plotted as a function of K+ ion concentration for KCl and K2ZnCl4 aqueous solutions. (From Jimbo et al., 1987.)

2. Supercontinuum Generation in Condensed Matter

Figure 2.23. (continued)

71

72

Q.Z. Wang, P.P. Ho, and R.R. Alfano Table 2.3. Enhancement of the supercontinuum and optical Kerr effects signals in saturated aqueous solutions at 20°C.a Signal

K2ZnCl4 (1.9 M)

KCl (4.0 M)

ZnCl2 (10.6 M)

GSPM (570) GSPM (500) GKerr

11 ± 1 9.5 ± 2.5 16 ± 1

5.6 ± 0.9 4.9 ± 0.2 6.1 ± 1.4

6.6 ± 0.4 4.3 ± 0.5 35 ± 9

GSPM(l) = [ISPM(l)/Ilaser(530 nm)]solution/[ISPM(l)/Ilaser (530 nm)]water and GKerr = [IKerr(solution)]/[IKerr (water)]. a

produced the greatest increase in the supercontinuum. Although ZnCl2 generated the largest enhancement of the optical Kerr effect, it did not play an important role in the enhancement of the ultrafast supercontinuum (a possible reason for this is discussed below). The optical Kerr effect signal from saturated solutions of ZnCl2 was about 2 to 3 times greater than that from saturated solutions of K2ZnCl4. The enhancement of the optical nonlinearity of water by the addition of cations can be explained by the cations’ disruption of the tetrahedral hydrogen-bonded water structures and their formation of hydrated units (Walrafen, 1972). Since the nonlinear index n2 is proportional to the number density of molecules, hydration increases the number density of water molecules and thereby increases n2. The ratio of the hydration numbers of Zn2+ and K+ has been estimated from measurements of GKerr and compared with their values based on ionic mobility measurements. At the same concentration of KCl and ZnCl2 acqueous solution, (GKerr generated by ZnCl2 solution)/(GKerr generated by KCl solution) = [N(Zn2+)/N(K+)]2 ~ 2.6, where N(Zn2+) ~ 11.2 ± 1.3 and N(K+) ~ 7 ± 1 represent the hydration numbers for the Zn2+ and K+ cations, respectively. The calculation of the hydration number of N(Zn2+)/N(K+) ~ 1.5 is in good agreement with the Kerr non-linearity measurements displayed in Table 2.3. In addition, from our previous measurements and discussions of nonlinear processes in mixed binary liquids (Ho and Alfano, 1978), the total optical nonlinearity of a mixture modeled from a generalized Langevin equation was determined by the coupled interactions of solute-solute, solute-solvent, and solvent-solvent molecules. The high salt solution concentration may contribute additional optical nonlinearity to the water owing to the distortion from the salt ions and the salt-water molecular interactions. The finding that Zn2+ cations increased GKerr more than GSPM is consistent with the hydration picture. The transmitted signal of the OKG depends on Dn, while the ultrafast supercontinuum signal is determined by ∂n/∂t. The ultrafast supercontinuum also depends on the response time of the hydrated units. Since the Zn2+ hydrated units are larger than those of K+, the response time will be longer. These effects will be reduced for longer pulses. Two addi-

2. Supercontinuum Generation in Condensed Matter

73

tional factors may contribute to part of the small discrepancy between GSPM and GKerr for ZnCl2. The first one is related to the mechanism of dn generation in which c1111 is involved in the generation of SPM while the difference c1111 - c1112 is responsible for the optical Kerr effect. The second is the possible dispersion of n2 because of the difference in wavelength between the exciting beams of the ultrafast supercontinuum and the optical Kerr effects. The optical Kerr effect is enhanced 35 times by using ZnCl2 as a solute, and the ultrafast supercontinuum is enhanced about 10 times by using K2ZnCl4 as a solute. The enhancement of the optical nonlinearity has been attributed to an increase in the number density of water molecules owing to hydration and the coupled interactions of solute and solvent molecules. Addition of ions can be used to increase n2 for SPM generation and gating.

11. Temporal Behavior of SPM In addition to spectral features, the temporal properties of the supercontinuum light source are important for understanding the generation and compression processes. In this section, the local generation, propagation, and pulse duration reduction of SPM are discussed.

11.1 Temporal Distribution of SPM In Section 2, using the stationary phase method, it was described theoretically that the Stokes and anti-Stokes frequencies should appear at welldefined locations in time within leading and trailing edges of the pump pulse profile (Alfano, 1972). Theoretical analyses by Stolen and Lin (1978) and Yang and Shen (1984) obtained similar conclusions. Passing an 80-fs laser pulse through a 500-mm-thick ethylene glycol jet stream, the pulse duration of the spectrum in time was measured by the autocorrelation method (Fork et al., 1983). These results supported the SPM mechanism for supercontinuum generation. In the following, the measurements of the distribution of various wavelengths for the supercontinuum generated in CCl4 by intense 8-ps laser pulses (Li et al., 1986) are presented. Reduction of the pulse duration using the SPM principle is discussed in Section 10.3. The incident 530-nm laser pulse temporal profile is shown in Figure 2.24. The pulse shape can be fitted with a Gaussian distribution with duration t(FWHM) = 8 ps. The spectral and temporal distributions of the supercontinuum pulse were obtained by measuring the time difference using a streak camera. The measured results are shown as circles in Figure 2.25. Each data point corresponds to an average of about six laser shots. The observation is consistent with the SPM and group velocity dispersion. To determine the temporal distribution of the wavelengths generated within a supercontinuum, the group velocity dispersion effect (Topp and Orner, 1975) in CCl4

74

Q.Z. Wang, P.P. Ho, and R.R. Alfano

Figure 2.24. Temporal profile of a 530-nm incident laser pulse measured by a 2-ps-resolution streak camera. The dashed line is a theoretical fit to an 8-ps EWHM Gaussian pulse. (From Li et al., 1986.)

was corrected. Results corrected for both the optical delay in the added filters and the group velocity are displayed as triangles in Figure 2.25.The salient feature of Figure 2.25 indicates that the Stokes wavelengths of the continuum lead the anti-Stokes wavelengths. Using the stationary phase SPM method [Eq. (6)], the generated instantaneous frequency w of the supercontinuum can be expressed by w (t) - w L = -(w L l c)∂ (Dn) ∂ t ,

(33)

where wL is the incident laser angular frequency, l is the length of the sample, and Dn is the induced nonlinear refractive index n2E2. A theoretical calculated curve for the sweep is displayed in Figure 2.26 by choosing appropriate parameters to fit the experimental data of Figure 2.25. An excellent fit using a stationary phase model up to maximum sweep demonstrates that the generation mechanism of the temporal distribution of the supercontinuum arises from the SPM. During the SPM process, a wavelength occurs at a welldefined time within the pulse. The above analysis will be supported by the additional experimental evidence for SPM described in Section 10.3 (see Figure 2.29).

2. Supercontinuum Generation in Condensed Matter

75

Figure 2.25. Measured supercontinuum temporal distribution at different wavelengths: (o) data points with correction of the optical path in filters; (D) data points with correction of both the optical path in filters and group velocity dispersion in liquid. (From Li et al., 1986.)

11.2 Local Generation and Propagation The dominant mechanisms responsible for the generation of the ultrafast supercontinuum as mentioned in Sections 1 and 2 are SPM, FPPG, XPM, and SRS. In the SPM process, a newly generated wavelength could have bandwidth-limited duration at a well-defined time location (Alfano, 1972) in the pulse envelope. In the FPPG and SRS processes, the duration of the supercontinuum pulse could be shorter than the pump pulse duration due to the high gain about the peak of the pulse. In either case, the supercontinuum pulse will be shorter than the incident pulse at the local spatial point of generation. These pulses will be broadened in time due to the group velocity dispersion in condensed matter (Ho et al., 1987).

76

Q.Z. Wang, P.P. Ho, and R.R. Alfano

Figure 2.26. Comparison of the measured temporal distribution of supercontinuum with the SPM model. (From Li et al., 1986.)

Typical data on the time delay of 10-nm-bandwith pulses centered at 530, 650, and 450 nm wavelengths of the supercontinuum generated from a 20-cm-long cell filled with CCl4 are displayed in Figure 2.27. The peak locations of 530, 650, and 450 nm are –49, –63, and –30 ps, respectively. The salient features in Figure 2.27 (Ho et al., 1987) indicate that the duration of all 10-nm-band supercontinuum pulses is only 6 ps, which is shorter than the incident pulse of 8 ps, the Stokes side (650 nm) of the supercontinuum pulse travels ahead of the pumping 530 nm by 14 ps, and the anti-Stokes side (450 nm) of the supercontinuum pulse lags the 530 nm by 10 ps. If the supercontinuum could be generated throughout the entire length of the sample, the Stokes side supercontinuum pulse generated by the 530-nm incident laser pulse at z = 0 cm of the sample would be ahead of the 530-nm incident pulse after propagating through the length of the sample. Over this path, 530 nm could continuously generate the supercontinuum pulse. Thus, the Stokes side supercontinuum generated at the end of the sample coincides in time with the 530-nm incident pulse. In this manner, a supercontinuum

2. Supercontinuum Generation in Condensed Matter

77

Figure 2.27. Temporal profiles and pulse locations of a selected 10-nm band of a supercontinuum pulse at different wavelengths propagated through a 20-cm-long CCl4 cell: (a) l = 530 nm; (b) l = 650 nm; (c) l = 450 nm. Filter effects were compensated. (From Ho et al., 1987.)

pulse centered at a particular Stokes frequency could have a pulse greater than the incident pulse extending in time from the energing of the 530-nm pulse to the position where the Stokes frequency was originally produced at z ~ 0 cm. From a similar consideration, the anti-Stokes side supercontinuum pulse would also be broadened. However, no slow asymmetric tail for the Stokes pulse or rise for the anti-Stokes pulse is displayed in Figure 2.27. These observation suggest the local generation of supercontinuum pulses. A model to describe the generation and propagation features of the supercontinuum pulse has been formulated based on local generation. The time delay of Stokes and anti-Stokes supercontinuum pulses relative to the 530-nm pump pulse is accounted for by the filaments formed ~5 cm from the sample cell entrance window. The 5-cm location is calculated from data in Figure 2.27 by using the equation. 1 ˆ Ê 1 T530 - Tsupercon. = DxÁ ˜, Ë v530 vsupercon. ¯

(34)

where Dx is the total length of supercontinuum pulse travel in CCl4 after the generation. T350 and Tsupercon. are the 530-nm and supercontinuum pulse peak

78

Q.Z. Wang, P.P. Ho, and R.R. Alfano

time locations in Figure 2.27, and v530 and vsupercon. are the group velocities of the 530-nm and supercontinuum pulses, respectively. The duration of the supercontinuum pulse right at the generation location is either limited by the bandwidth of the measurement from the SPM process or shortened by the parametric generation process. In either cases, a 10-nmbandwidth supercontinuum pulse will have a shorter duration than the incident pulse. After being generated, each of these 10-nm-bandwidth supercontinuum pulses will travel through the rest of the sample and will continuously generated by the incident 530 nm over a certain interaction length before these two pulses walk off. The interaction length can be calculated as (Alfano, 1972). l =t

v530vsupercon. , v530 - vsupercon.

(35)

where l is the interaction length over the pump and the supercontinuum pulses stay spatially coincident by less than the duration (FWHM) of the incident pump pulse, and t is the duration of the supercontinuum pulse envelope. From Eq. (35), one can estimate the interaction length from the measured t of the supercontinuum pulse. Using parameters t = 6 ps, v530 = c/1.4868, and vsupercon. = c/1.4656, the interaction length l = 8.45 cm is calculated. This length agrees well with the measured beam waist length of 8 cm for the pump pulse in CCl4. Since no long tails were observed from the supercontinuum pulses to the dispersion delay times of the Stokes and anti-Stokes supercontinuum pulses, the supercontinuum was not generated over the entire length of 20 cm but only over 1 to 9 cm. This length is equivalent to the beam waist length of the laser in CCl4. The length of the local SPM generation over a distance of 8.45 cm yields a possible explanation for the 6-ps supercontinuum pulse duration. In addition, a pulse broadening of 0.3 ps calculated from the group velocity dispersion of a 10-nm band at 650-nm supercontinuum traveling over 20 cm of liquid CCl4 is negligible in this case. Therefore, the SPM pulses have shorter durations than the pump pulse and were generated over local spatial domains in the liquid cell.

11.3 SPM Pulse Duration Reduction The principle behind the pulse narrowing based on the spectral temporal distribution of the SPM spectrally broadened in time within the pulse is described in Sections 2 and 10.1. At each time t within the pulse there is a frequency w(t). When a pulse undergoes SPM, the changes in the optical carrier frequency within the temporal profile are greatest on the rising and falling edges, where the frequency is decreased and increased, respectively. Near the peak of the profile, and in the far leading and trailing wings, the carrier frequency structure is essentially unchanged. The maximum frequency shift is proportional to the intensity gradient on the sides of the pulse, and

2. Supercontinuum Generation in Condensed Matter

79

Figure 2.28. Streak camera temporal profile of the 25-ps, 530-nm incident laser pulse and 10-nm-bandwidth pulse at 580 nm. The 3-ps pulse was obtained by spectral filtering a SPM frequency continuum generated in D2O. (From Dorsinville et al., 1987.)

this determines the position of the outer lobes of the power spectrum. If these are then attenuated by a spectral window of suitably chosen width, the wings of the profile where the high- and low-frequency components are chiefly concentrated will be depressed, while the central peak will be largely unaffected. The overall effect is to create a pulse that is significantly narrower in time than the original pulse duration. A file can be used to select a narrow portion of the pulse, giving rise to a narrower pulse in time. A threefold shortening of 80-ps pulses to 30 ps from an Nd : YAG laser broadened from 0.3 to 4 Å after propagation through 125 m of optical fiber with a monochromator as a spectral window was demonstrated using this technique (Gomes et al., 1986). The measurements of pulses at different wave-lengths of the frequency sweep of supercontinuum pulses generated by 8-ps laser pulses propagating in CCl4 show that the continuum pulses have a shorter duration (~6 ps) than the pumping pulses (Li et al., 1986). A major advance occurred when a 25-ps laser pulse was focused into a 5-cm-long cell filled with D2O. A continuum was produced. Using 10-nmbandwidth narrowband filters, tunable pulses of less than 3 ps in the spectral range from 480 to 590 nm (Figure 2.28) were produced (Dorsinville et al., 1987). To identify the SPM generation mechanism, the temporal distribution of the continuum spectrum was determined by measuring the time delay between the continuum and a reference beam at different wavelengths using a streak camera. The results are displayed in Figure 2.29, which is similar to data displayed in Figure 2.25. The time delay was ~22 ps for a 140-nm change in wavelength; as predicted by the SPM mechanism, the Stokes wavelength led the anti-Stokes wavelength (Alfano, 1972). The delay due to group velocity over a 5-cm D2O cell for the 140-nm wavelength change is less than 3 ps.

80

Q.Z. Wang, P.P. Ho, and R.R. Alfano Figure 2.29. Continuum temporal distribution at different wavelengths. Horizontal error bars correspond to 10-nm band-widths of the filter. (From Dorsinville et al., 1987.)

The remaining 18 ps is well accounted for by the SPM mechanism using a 25-ps (FWHM) pulse and the stationary phase method (Alfano, 1972). Furthermore, a 10-nm selected region in the temporal distribution curve corresponds to an ~2.6-ps width matching the measured pulse duration (Figure 2.28). This observation suggests that by using narrower bandwidth filters the pulse duration can be shortened to the uncertainty limit.

12. Higher-Order Effects on Self-Phase Modulation A complete description of SPM-generated spectral broadening should take into account higher-order effects such as self-focusing, group velocity dispersion, self-steepening, and initial pulse chirping. Some of these effects are described by Suydam (Chapter 6), Shen (Chapter 1), and Agrawal (Chapter 3). These effects will influence the observed spectral profiles.

12.1 Self-Focusing In the earliest experiments using picosecond pulses, the supercontinuum pulses were often generated in small-scale filaments resulting from the selffocusing of intense laser beams (Alfano, 1972). Self-focusing arises from the radial dependence of the nonlinear refractive index n(r) = n0 + n2E2(r) (Shen, 1984; Auston, 1977). It has been observed in many liquids, bulk materials (Shen, 1984), and optical fibers (Baldeck et al., 1987a). It effects on the continuum pulse generation can be viewed as good and bad. On the one hand,

2. Supercontinuum Generation in Condensed Matter

81

it facilitates the spectral broadening by concentrating the laser beam energy. On the other hand, self-focusing is a random and unstable phenomenon that is not controllable. Femtosecond supercontinua are generated with thinner samples than picosecond supercontinua, so it can reduce but not totally eliminate self-focusing effects.

12.2 Dispersion Group velocity dispersion (GVD) arises from the frequency dependence of the refractive index. These effects are described by Agrawal (Chapter 3). The firstorder GVD term leads to a symmetric temporal broadening (Marcuse, 1980). A typical value for the broadening rate arising from ∂ 2k/∂w 2 is 500 fs/m · nm (in silica at 532 nm). In the case of supercontinuum generation, spectral widths are generally large (several hundred nanometers), but interaction lengths are usually small (<1 cm). Therefore, the temporal broadening arising from GVD is often negligible for picosecond pulses but is important for femtosecond pulses. Limitations on the spectral extent of supercontinuum generation are also related to GVD. Although the spectral broadening should increase linearly with the medium length (i.e., Dw(z)max = w 0n2a2z/c Dt), it quickly reaches a maximum as shown in Figure 2.10. This is because GVD, which is large for pulses having SPM-broadened spectra, reduces the pulse peak power a2 and broadens the pulse duration Dt. As shown in Figure 2.25, the linear chirp parameter is decreased by the GVD chirp in the normal dispersion regime. This effect is used to linearize chirp in the pulse compression technique. The second-order term ∂ 3k/∂w 3 has been found to be responsible for asymmetric distortion of temporal shapes and modulation of pulse propagation in the lower region of the optical fiber (Agrawal and Potasek, 1986). Since the spectra of supercontinuum pulses are exceptionally broad, this term should also lead to asymmetric distortions of temporal and spectral shapes of supercontinuum pulses generated in thick samples. These effects have been observed. In multimode optical fibers, the mode dispersion dominates and causes distortion of the temporal shapes. This in turn yields asymmetric spectral broadening (Wang et al., 1988).

12.3 Self-Steepening Pulse shapes and spectra of intense supercontinuum pulses have been found to be asymmetric (De Martini et al., 1967). There are two potential sources of asymmetric broadening in supercontinuum generation. The first one is the second-order GVD term. The second one is self-steepening, which is intrinsic to the SPM process and occurs even in nondispersion media. Details of the effects of self-steepening can be found in Suydam (Chapter 6), Shen and Yang (Chapter 1), and Manassah (Chapter 5). Because of the intensity and time dependence of the refractive index, n = n0 + n2E(t)2, the supercontinuum pulse peak sees a higher refractive index than its edges. Because v = c/n, the pulse peak travels slower than the leading and

82

Q.Z. Wang, P.P. Ho, and R.R. Alfano Figure 2.30. Influence of initial pulse chirping on SPM-broadened spectra in optical fibers. Peak power = 1000 W. (a) C = 50; (b) C = 0; (c) C = -50 [see Eq. (17)]. (From Baldeck et al., 1987b.)

trailing edges. This results in a sharpened trailing edge. Self-steepening occurs and more blue-shifted frequencies (sharp trailing edge) are generated than red frequencies. Several theoretical approaches have given approximate solutions for the electric field envelope distorted by self-steepening and asymmetric spectral extent. Actual self-steepening effects have not been observed in the time domain.

12.4 Initial Pulse Chirping Most femtosecond and picosecond pulses are generated with initial chirps. Chirps arise mainly from GVD and SPM in the laser cavity. As shown in Figure 2.30, the spectral broadening is reduced for positive chirps and enhanced for negative chirps in the normal dispersion regime. The spectral distribution of SPM is also affected by the initial chirp.

13. Overview Supercontinuum generation is the generation of bursts of “white” light, which can be obtained by passing intense picosecond or femtosecond pulses through various materials. Because of the nonlinear response of the medium,

2. Supercontinuum Generation in Condensed Matter

83

the pulse envelope yields a phase modulation that initiates the wide frequency broadening (up to 10,000 cm-1). The phase modulation can be generated by the pulse itself, a copropagating pump pulse, or the copropagating stronger pulse. These different configurations are called self-phase modulation (SPM), induced-phase modulation (IPM), and cross-phase modulation (XPM), respectively. The SPM process for supercontinuum generation in various materials was reviewed in this chapter. This latter two processes are closely related to each other and are described by Baldeck et al. (Chapter 4), Agrawal (Chapter 3), and Manassah (Chapter 5). Using an 8-ps laser at 530 nm, typical Stokes sweeps were 4400 cm-1 in a calcite crystal of length 4 cm, 3900 cm-1 in a quartz crystal of length 4.5 cm, 1100 cm-1 in extra-dense flint glass of length 7.55 cm, 3900 cm-1 in NaCl of length 4.7 cm, and 4200 cm-1 in both BK-7 and LBC-1 glasses of length 8.9 cm. Sweeps on the anti-Stokes side were typically 6100 cm-1 in calcite, 5500 cm-1 in quartz, 7300 cm-1 in NaCl, and 7400 cm-1 in BK-7 and LBC-1 glasses. An infrared supercontinuum spanning the range from 3 to 14 mm can be obtained by passing an intense laser pulse generated from a CO2 laser through GaAs, AgBr, ZnSe, and CdS crystals. Near- and medium-infrared spectral sweeps of 3200 cm-1 on the Stokes side and 4900 cm-1 on the antiStokes side can be realized by passing a strong 1.06-mm pulse through a KBr crystal of length 10 cm. Sweeps on the order of 1000 cm-1 are observed to both the red and blue sides of 530 nm in liquid argon. Similar spectral sweeps are observed in liquid and solid krypton arising from electronic mechanism for SPM. Using a picosecond laser train of wavelength 530 nm, the spectra were broadened up to 3000 cm-1 to either side of the laser frequency in a 5-cm-long magnetic KNiF3 single crystal. Production of SPM near electronic levels of PrF3 crystal and enhancement of supercontinuum in water by addition of Zn2+ and K+ cations have been also discussed. The temporal properties of supercontinuum pulses have been described. Higher-order effects on SPM arising from dispersion, self-focusing, self-steepening, and initial pulse chirping were briefly described. SPM will continue to be an important nonlinear process in science and technology and has been one of the most important ultrafast nonlinear optical processes for more than 20 years since the advent of ultrashort laser pulses!

References Agrawal, G.P. and M.J. Potasek (1986) Nonlinear pulse distortion in single-mode optical fibers at the zero-dispersion wavelength. Phys. Rev. A 33, 1765–1776. Alfano, R.R. (1972) Interaction of picosecond laser pulses with matter. GTE Technical Report TR 72–330. Published as Ph.D. thesis at New York University, 1972. Alfano, R.R. (1986) The ultrafast supercontinuum laser source. Proc. International Conference Laser ’85. STS Press, McLean, Virigina, pp. 110–122. Alfano, R.R. and P.P. Ho (1988) Self-, cross-, and induced-phase modulations of ultrashort laser pulse propagation. IEEE J. Quantum Electron. QE-24, 351–363.

84

Q.Z. Wang, P.P. Ho, and R.R. Alfano

Alfano, R.R. and S.L. Shapiro (1970a) Emission in the region 4000–7000 A via fourphoton coupling in glass. Phys. Rev. Lett. 24, 584–587; Observation of self-phase modulation and small scale filaments in crystals and glasses. Phys. Rev. Lett. 24, 592–594; Direct distortion of electronic clouds of rare-gas atoms in intense electric fields. Phys. Rev. Lett. 24, 1219–1222. Alfano, R.R. and S.L. Shapiro (1970b) Picosecond spectroscopy using the inverse Raman effect. Chem. Phys. Lett. 8, 631–633. Alfano, R.R., L. Hope, and S. Shapiro (1972) Electronic mechanism for production of self-phase modulation. Phys. Rev. A 6, 433–438. Alfano, R.R., J. Gersten, G. Zawadzkas, and N. Tzoar (1974) Self-phase-modulation near the electronic resonances of a crystal. Phys. Rev. A 10, 698–708. Alfano, R.R., P. Ho, P. Fleury, and H. Guggeneheim (1976) Nonlinear optical effects in antiferromagnetic KNiF3. Opt. Commun. 19, 261–264. Alfano, R.R., Q. Li, T. Jimbo, J. Manassah, and P. Ho (1986) Induced spectral broadening of a weak picosecond pulse in glass produced by an intense ps pulse. Opt. Lett. 11, 626–628. Alfano, R.R., Q.Z. Wang, T. Jimbo, and P.P. Ho (1987) Induced spectral broadening about a second harmonic generated by an intense primary ultrashort laser pulse in ZnSe crystals. Phys. Rev. A 35, 459–462. Alfano, R.R., Q.Z. Wang, D. Ji, and P.P. Ho (1989) Harmonic cross-phasemodulation in ZnSe. App. Phys. Lett, 54, 111–113. Anderson, D. and M. Lisak (1983) Nonlinear asymmetric self-phase modulation and self-steepening of pulses in long optical waveguides. Phys. Rev. A 27, 1393–1398. Auston, D.H. (1977) In Ultrafast Light Pulses, S.L. Shapiro, ed., Springer, Verlag, New York. Baldeck, P.L., F. Raccah, and R.R. Alfano (1987a) Observation of self-focusing in optical fibers with picosecond pulses. Opt. Lett. 12, 588–589. Baldeck, P.L., P.P. Ho, and R.R. Alfano (1987b) Effects of self-, induced-, and crossphase modulations on the generation of ps and fs white light supercontinuum. Rev. Phys. Appl. 22, 1877–1894. Bloembergen, N. and P. Lallemand (1966) Complex intensity dependent index of refraction frequency broadening of stimulated Raman lines and stimulated Rayleigh scattering. Phys. Rev. Lett. 16, 81–84. Bourkoff, E., W. Zhao, and R.I. Joseph (1987) Evolution of femtosecond pulses in single-mode fibers having higher-order nonlinearity and dispersion. Opt. Lett, 12, 272–274. Brewer, R.G. (1967) Frequency shifts in self-focused light. Phys. Rev. Lett. 19, 8–10. Brewer, R.G. and C.H. Lee (1968) Self-trapping with picosecond light pulses. Phys. Rev. Lett. 21, 267–270. Busch, G.E., R.P. Jones, and P.M. Rentzepis (1973) Picosecond spectroscopy using a picosecond continuum. Chem. Phys. Lett. 18, 178–185. Chinn, S.R., H. Zeiger, and J. O’Connor (1971) Two-magnon Raman scattering and exchange interactions in antiferromagnetic KNiF3 and K2NiF4 and ferrimagnetic RbNiF3. Phys. Rev. B3, 1709–1735. Corkum, P., P. Ho, R. Alfano, and J. Manassah (1985) Generation of infrared supercontinuum covering 3–14 mm in dielectrics and semiconductors. Opt. Lett. 10, 624–626. Corkum, P.B., C. Rolland, and T. Rao (1986) Supercontinuum generation in gases. Phys. Rev. Lett. 57, 2268–2271.

2. Supercontinuum Generation in Condensed Matter

85

Cornelius, P. and L. Harris (1981) Role of self-phase modulation in stimulated Raman scattering from more than one mode. Opt. Lett. 6, 129–131. DeMartini, F., C.H. Townes, T.K. Gustafson, and P.L. Kelly (1967) Self-steepening of light pulses. Phys. Rev. 164, 312–322. Dorsinville, R., P. Delfyett, and R.R. Alfano (1987) Generation of 3 ps pulses by spectral selection of the supercontinuum generated by a 30 ps second harmonic Nd: YAG laser pulse in a liquid. Appl. Opt. 27, 16–18. Fisher, R.A. and W. Bischel (1975) Numerical studies of the interplay wave laser pulse. J. Appl. Phys. 46, 4921–4934. Fisher, R.A., B. Suydam, and D. Yevich (1983) Optical phase conjugation for time domain undoing of dispersion self-phase modulation effects. Opt. Lett. 8, 611–613. Fleury, P.A., W. Hayes, and H.J. Guggenheim (1975) Magnetic scattering of light in K(NiMg)F3. J. Phys. C: Solid State 8, 2183–2189. Fork, R.L., C.V. Shank, C. Hirliman, R. Yen, and J. Tomlinson (1983) Femtosecond white-light continuum pulse. Opt. Lett. 8, 1–3. Fork, R.L., C.H. Brito Cruz, P.C. Becker, and C.V. Shank (1987) Compression of optical pulses to six femtoseconds by using cubic phase compensation. Opt. Lett. 12, 483–485. Gersten, J., R. Alfano, and M. Belic (1980) Combined stimulated Raman scattering in fibers. Phys. Rev. A 21, 1222–1224. Girodmaine, J.A. (1962) Mixing of light beams in crystals. Phys. Rev. Lett. 8, 19–20. Glownia, J., G. Arjavalingam, P. Sorokin, and J. Rothenberg (1986) Amplification of 350-fs pulses in XeCl excimer gain modules. Opt. Lett. 11, 79–81. Goldberg, L. (1982) Broadband CARS probe using the picosecond continua. In Ultrafast Phenomena III. Springer-Verlag, New York, pp. 94–97. Gomes, A.S.L., A.S. Gouveia-Neto, J.R. Taylor, H. Avramopoulos, and G.H.C. New (1986) Optical pulse narrowing by the spectral windowing of self-phase modulated picosecond pulses. Opt. Commun. 59, 399. Gustafson, T.K., I.P. Taran, H.A. Haus, J.R. Lifisitz, and P.L. Kelly (1969) Selfmodulation, self-steepening, and spectral development of light in small-scale trapped filaments. Phys. Rev 177, 306–313. Gustafson, T.K., J. Taran, P. Kelley, and R. Chiao (1970) Self-modulation of picosecond pulse in electro-optical crystals. Opt. Commun. 2, 17–21. Hellwarth, R.W. (1970) Theory of molecular light scattering spectra using lineardipole approximation. J. Chem. Phys. 52, 2128–2138. Hellwarth, R.W., J. Cherlow, and T.T. Yang (1975) Origin and frequency dependence of nonlinear optical susceptibilities of glasses. Phys. Rev. B 11, 964–967. Heritage, J., A. Weiner, and P. Thurston (1985) Picasecond pulse shaping by spectral phase and amplitude manipulation. Opt. Lett. 10, 609–611. Ho, P.P. and R.R. Alfano (1978) Coupled molecular reorientational relaxation kinetics in mixed binary liquids directly measured by picosecond laser techniques. J. Chem. Phys. 68, 4551–4563. Ho, P.P. and R.R. Alfano (1979) Optical Kerr effect in liquids. Phys. Rev. A 20, 2170–4564. Ho, P.P., Q.X. Li, T. Jimbo, Y.L. Ku, and R.R. Alfano (1987) Supercontinuum pulse generation and propagation in a liqud carbon tetrachloride. Appl. Opt. 26, 2700–2702. Ishida, Y., K. Naganuma, T. Yagima, and C. Lin (1984) Ultrafast self-phase modulation in a colliding pulse mode-locking ring dye laser. In Ultrafast Phenomena IV. Springer-Verlag, New York, pp. 69–71.

86

Q.Z. Wang, P.P. Ho, and R.R. Alfano

Jimbo, T., V.L. Caplan, Q.X. Li, Q.Z. Wang, P.P. Ho, and R.R. Alfano (1987) Enhancement of ultrafast supercontinuum generation in water by addition of Zn2+ and K+ cations. Opt. Lett. 12, 477–479. Johnson, A., R. Stolen, and W. Simpson (1986) The observation of chirped stimulated Raman scattering light in fibers. In Ultrafast Phenomena V.G.R. Fleming and A.E. Siegman ed. Springer-Verlag, New York, pp. 160–163. Jones, W.J. and B.P. Stoicheff (1964) Inverse Raman spectra: induced absorption at optical frequencies. Phys. Rev. Lett. 13, 657–659. Knox, K., R.G. Shulman, and S. Sugano (1963) Covalency effects in KNiF3. II. Optical studies. Phys. Rev. 130, 512–516. Knox, W., R. Fork, M. Dower, R. Stolen, and C. Shank (1985) Optical pulse compression to 8-fs at 5-kHz repetition rate. Appl. Phys. Lett. 46, 1120–1121. Kobayashi, T. (1979) Broadband picosecond light generation in phosphoric acid by a mode-locked laser. Opt. Commun. 28, 147–149. Lallemand, P. (1966) Temperature variation of the width of stimulated Raman lines in liquids. Appl. Phys. Lett. 8, 276–277. Levenson, M.D. and N. Bloembergen (1974) Dispersion of the nonlinear optical susceptibility tensor in centrosymmetric media. Phys. Rev. B 10, 4447–4463. L, Q.X., T. Jimbo, P.P. Ho, and R.R. Alfano (1986) Temporal distribution of picosecond super-continuum generated in a liquid measured by a streak camera. Appl. Opt. 25, 1869–1871. Lozobkin, V., A. Malytin, and A. Prohorov (1970) Phase self-modulation of Nd: glass radiation with mode-locking. JETP Lett. 12, 150–152. Magde, D. and M.W. Windsor (1974) Picosecond flash photolysis and spectroscopy: 3,3’-diethyloxadicarbocyanine iodide (DODCI). Chem. Phys. Lett. 27, 31–36. Manassah, J.T., P.P. Ho, A. Katz, and R.R. Alfano (1984) Ultrafast supercontinuum laser source. Photonics Spectra 18 November, 53–59. Manassah, J.T, R.R., Alfano and M. Mustafa (1985a) Spectral distribution of an ultrashort supercontinuum laser source. Phys. Lett. A 107, 305–309. Manassah, J.T., M. Mustafa, R. Alfano, and P. Ho (1985b) Induced supercontinuum and steepening of an ultrafast laser pulse. Phys. Lett. 113A, 242–247. Manassah, J.T., M. Mustafa, R.R. Alfano, and P.P. Ho (1986) Spectral extent and pulse shape of the supercontinuum for ultrashort laser pulse. IEEE J. Quantum Electron. QE-22, 197–204. Marcuse, D. (1980) Pulse distortion in single-mode optical fibers Appl. Opt. 19, 1653–1660. Masuhara, H., H. Miyasaka, A. Karen, N. Mataga, and Y. Tsuchiya (1983) Temporal characteristics of picosecond continuum as revealed by two-dimensional analysis of streak images. Opt. Commun. 4, 426. McTague, J., P., Fleury, and D. DuPre (1969) Intermolecular light scattering in liquids. Phys. Rev. 188, 303–308. Nakashima, N. and N. Mataga (1975) Picosecond flash photolysis and transient spectral measurements over the entire visible, near ultraviolet and near infrared regions. Chem. Phys. Lett. 35, 487–492. Patel, C.K.N. and E.D. Shaw (1971) Tunable stimulated Raman scattering from mobile carriers in semiconductors. Phys. Rev. B 3, 1279–1295. Penzokfer, A., A. Laubereau, and W. Kasier (1973) Stimulated short-wave radiation due to single frequency resonances of c(3). Phys. Rev. Lett. 31 863–866.

2. Supercontinuum Generation in Condensed Matter

87

Potasek, M.J., G.P. Agrawal, and S.C. Pinault (1986) Analytical and numerical study of pulse broadening in nonlinear dispersive optical fibers. J. Opt. Soc. Am. B 3, 205–211. Shank, C. (1983) Measurement of ultrafast phenomena in the femtosecond domain. Science 219, 1027. Shank, C.V., R.L. Fork, R. Yen, and R.H. Stolen (1982) Compression of femtosecond optical pulses. Appl. Phys. Lett. 40, 761–763. Sharma, D.K., R.W. Yid, D.F. Williams, S.E. Sugamori, and L.L.T. Bradley (1976) Generation of an intense picosecond continuum in D2O by a single picosecond 1.06 m pulse. Chem. Phys. Lett. 41, 460–465. Shen, Y.R. (1966) Electrostriction, optical Kerr effect and self-focusing of laser beams. Phys. Lett. 20, 378. Shen Y.R. (1984) The Principles of Nonlinear Optics. Wiley, New York. Shimizu, F. (1967) Frequency broadening in liquids by a short light pulse. Phys. Rev. Lett. 19, 1097–1100. Stolen, R.H. and A.M. Johnson (1986) The effect of pulse walkoff on stimulated Raman scattering in optical fibers. IEEE J. Quantum Electron. QE-22, 2154–2160. Stolen, R.H. and C. Lin (1978) Self-phase modulation in silica optical fibers. Phys. Rev. A 17, 1448–1453. Topp, M.R. and G.C. Orner (1975) Group velocity dispersion effects in picosecond spectroscopy. Opt. Commun. 13, 276. Tzoar, N. and M. Jain (1981) Self-phase modulation in long-geometry waveguide. Phys. Rev. A 23, 1266–1270. Walrafen, G.E. (1972) Stimulated Raman scattering and the mixture model of water structure. Adv. Mol. Relaxation Processes 3, 43–49. Wang, Q.Z., D. Ji, Lina Yang, P.P. Ho, and R.R. Alfano (1989) Self-phase modulation in multimode optical fibers with modest high power. Produced by moderately high power picosecond pulses. Opt. Lett., in press. Yablonovitch, E. and N. Bloembergen (1972) Avalanche ionization of the limiting diameter of filaments induced by light pulses in transparent media. Phys. Rev. Lett. 29, 907–910. Yang, G. and Y.R. Shen (1984) Spectral broadening of ultrashort pulse in a nonlinear medium. Opt. Lett. 9, 510–512. Yu, W., R. Alfano, C.L. Sam, and R.J. Seymour (1975) Spectral broadening of picosecond 1.06 mm pulse in KBr. Opt. Commun. 14, 344.

Appendix: Nonlinear Wave Equation with Group Velocity Dispersion We start with Maxwell equations for the electric and magnetic fields E and H in Gaussian units 1 ∂B , c ∂t 1 ∂ D 4p —¥H= + J, c ∂t c —◊ D = 4pr, —◊ B = 0. —¥E=-

(A.1)

88

Q.Z. Wang, P.P. Ho, and R.R. Alfano

The helping equations are D = eE and B = mH, and J and r are the current and charge densities, respectively. For nonmagnetic material, B ª H. The refractive index of an isotropic material possessing nonlinearity can be written as n(w ) = [e (w )]

12

= n0 (w ) + n2 E , 2

(A.2)

where n0(w) is the linear refractive index and n2 the nonlinear refractive index. In the absence of sources, from Maxwell equations one can readily obtain the wave equation — 2 E(r,t) -

1 ∂2 2n0 n2 ∂ 2 2 DL (r,t) = E E(r,t) , 2 2 c ∂t c2 ∂t2

(

)

(A.3)

where DL(r, t) is the linear electric displacement vector. In obtaining the equation, we have used — ¥ (— ¥ E) = —(—·E) - —2E) ª -—2E and neglected the (n2)2 term. The electric field can be written as E(r,t) = F(x, y)E(z,t) ,

(A.4)

where F(x, y) is the transverse distribition function. Substitute Eq. (A.4) into the wave equation and averaging over transverse coordinates, we have ∂2 1 ∂2 2n0 n2 ∂ 2 2 ( ) ( ) E z , t D z , t = E(z,t) E(z,t). L 2 2 2 2 ∂z c ∂t c ∂t

(A.5)

We have neglected the ∂ 2/∂x2 and ∂ 2/∂y2 terms. The effective nonlinear refractive index n–2 is n2 =

Ú n F (x, y)dxdy ª 1 n . Ú F (x, y)dxdy 2 2

2

2

2

(A.6)

Using a plane wave approximation with (k0z - w0t) representation, a linearly polarized electric field propagating along z direction can be written as E(z,t) = eˆ A(z,t) exp[i (k0 z - w 0t)],

(A.7)

where ê is the unit vector of polarization of electric field, w0 the carrier frequency, k0 the carrier wave number, and A(z, t) the pulse envelope function. The form of DL(z, t) becomes +•

DL (z,t) = Ú n02 (w )E˜ (z,w ) exp(-iwt)dw , -•

(A.8)

where 1 E˜ (z,w ) = 2p

Ú

+•

-•

E (z,t) exp(iwt)dt.

(A.9)

2. Supercontinuum Generation in Condensed Matter

89

If the (w 0t - k0z) representation is used, one obtains sign changes in the final reduced wave equation. Using the foregoing equations, we can write the linear polarization term on the left-hand side of the one-dimensional wave equation as -

1 ∂2 DL (z,t) c2 ∂t2 1 +• = - 2 Ú (-w 2 )n02 (w )E˜ (z,w ) exp(-iwt)dw c -• 1 • • 2 k (w )A(z,t ¢) exp[iw (t ¢ - t)] exp[i (k0 z - w 0t ¢)]dwdt ¢. (A.10) = 2p Ú-• Ú-•

The derivation of the wave equation then proceeds by expanding k2(w) about the carrier frequency w0 in the form: k 2 (w ) ª k02 + 2k0 k0(1) (w - w 0 ) + k0 kP( 2) (w - w 0 ) + . . . , 2

(1) where k0 = k(w0) is the propagation constant, k0 =

∂k ∂w

(A.11)

is the inverse of w =w 0

∂ 2k is the inverse of group velocity of disper∂ w w =w0 sion. It is then possible to evaluate the integral of Eq. (A.10) by using the convenient delta function identities group velocity, and k0( 2) =

1 2p

Ú

•

exp[i (w - w 0 )(t ¢ - t)]dw = d (t ¢ - t)

(A.12)

(w - w 0 ) exp[i (w - w 0 )(t ¢ - t)]dw = id (1) (t ¢ - t) ,

(A.13)

(w - w 0 ) 2 exp[i (w - w 0 )(t ¢ - t)]dw = -d ( 2) (t ¢ - t) .

(A.14)

-•

as well as 1 2p

Ú

•

-•

and 1 2p

Ú

•

-•

In these relations, d (n)(t) is an nth-order derivative of the Dirac delta function, with the property that

Ú

•

-•

d ( n) (t - t0 ) f (t)dt =

d n f (t) ∂ t n t =t0

(A.15)

when applied to a function f(t). Substitute Eq. (A.11) into Eq. (A.10) and use Eq. (A.12) to (A.15), the second term on the left-hand side of Eq. (1.5) becomes

90

Q.Z. Wang, P.P. Ho, and R.R. Alfano

-

1 ∂2 DL (z,t) c2 ∂t2 1 • • 2 2 = k0 + 2k0 k0(1) (w - w 0 ) + k0 k0( 2) (w - w 0 ) A(z,t ¢) Ú Ú -• -• 2p ¥ exp[iw (t ¢ - t)] exp[i (k0 z - w 0t)]d w dt ¢

[

=Ú

•

-•

]

[k02d (t ¢ - t) + i 2k0 k0(1)d (1) (t ¢ - t) - k0 k0( 2)d ( 2) (t ¢ - t)]A(z,t ¢)

¥ exp[i (k0 z - w 0t)]dt ¢

∂A ∂ 2A˘ È = Ík02 A + i 2k0 k0( 2) - k0 k0( 2) 2 ˙ exp[i (k0 z - w 0t)]. ∂t ∂t ˚ Î

(A.16)

∂2 2 A(z, t) . 2 ∂t A(z, t), the first term on the left-hand side and the term on the right-hand side of Eq. (1.5) are simply Neglecting the second derivative of A(z, t) with respect to z and

∂ 2 E (z,t) È 2 ∂ A(z,t) ˘ ª Í-k0 A(z,t) + i 2k0 exp[i (k0 z - w 0t)], 2 ∂z ∂ z ˙˚ Î

(A.17)

2n0 n2 ∂ 2 n0 n2w 02 2 2 E EªA A exp[i (k0 z - w 0t)], 2 2 c ∂t c2

(A.18)

and

respectively. Inserting Eqs. (A.16) to (A.18) into Eq. (1.5), the wave equation for electric field reduces to the wave equation for the pulse envelope Ê ∂ A 1 ∂ A ˆ 1 ( 2) ∂ 2 A w 0 2 iÁ n2 A A = 0, + + ˜- k Ë ∂ z v g ∂ t ¯ 2 0 ∂ t 2 2c

(A.19)

where vg ∫ 1/k(1) 0 is the group velocity. In Eq. (A.19), the first two terms describe the envelope propagation at the group velocity vg; the third term determines the temporal pulse broadening due to group velocity dispersion; the fourth characterizes the second order of the nonlinear polarization, which is responsible for the self-phase modulation effect and spectral broadening. Neglecting the group velocity dispersion term in Eq. (A.19), we obtain ∂A 1 ∂A w0 2 + =i n2 A A. ∂ z vg ∂ t 2c This is Eq. (2.2).

(A.20)

3

Ultrashort Pulse Propagation in Nonlinear Dispersive Fibers Govind P. Agrawal

1. Introduction The use of silica fibers for transmission of optical pulses has become widespread, as is evident from the recent advances in optical fiber communications (Basch, 1986; Miller and Kaminow, 1988). For pulses not too short (pulse width >1 ns) and not too intense (peak power <10 mW), the fiber plays a passive role (except for energy loss) and acts as a transporter of optical pulses from one place to another without significantly affecting their shape or spectrum. However, as pulses become shorter and more intense, two physical mechanisms, chromatic dispersion and index nonlinearity, both intrinsic to the silica material, start to affect the pulse shape and spectrum during propagation. The fiber has found many novel and interesting applications in this active role. It has been used for pulse compression (Shank et al., 1982; Nikolaus and Grischkowsky, 1983), and pulses with durations as short as 6 fs have been produced (Fork et al., 1987). Such short pulses provide the capability for ultrafast continuum spectroscopy with a time resolution approximating that allowed by the transform-limited bandwidth (Shank et al., 1986). In the anomalous dispersion regime the fiber supports optical solitons resulting from a balance between the dispersive and nonlinear effects (Hasegawa and Tappert, 1973; Mollenauer et al., 1980). The solitons may be useful for high-speed optical communications over long distances (Hasegawa, 1983; Mollenauer et al., 1986). The soliton formation capacity of optical fibers has also been exploited to develop the soliton laser (Mollenauer and Stolen, 1984). An important application of optical fibers is in the field of ultrafast super-continuum generation (Alfano, 1985; Shank et al., 1986). Several nonlinear effects such as self-phase modulation (SPM), cross-phase modulation (XPM), and stimulated Raman scattering (SRS) can lead to an extensive spectral broadening of the incident pulse, resulting in an almost “white” spectrum (Alfano and Shapiro, 1970). Although SPM in optical fibers was studied nearly a decade ago (Stolen and Lin, 1978), it is only recently that the effect of XPM on propagation of ultrashort pulses in optical fibers has attracted

92

G.P. Agrawal

attention (Alfano et al., 1987; Islam et al., 1987; Agrawal, 1987; Schadt and Jaskorzynska, 1987; Baldeck et al., 1987). This chapter reviews how the nonlinear and dispersive effects in optical fibers influence the propagation characteristics of ultrashort pulses with widths in the picosecond range. In Section 2 we outline the derivation of the basic propagation equation satisfied by the amplitude of the pulse envelope. In the presence of SRS it becomes necessary to consider the coupled amplitude equations satisfied by the pump and Raman pulses. These equations include the effect of group velocity mismatch, group velocity dispersion, SPM, XPM, SRS gain, and pump depletion. Section 3 considers the propagation of a single pulse below the SRS threshold and identifies various propagation regimes. Sections 4 and 5 then consider the cases of normal dispersion and anomalous dispersion regimes separately. The effect of SRS on pulse propagation is studied in Section 6, where the results of numerical simulations are presented and compared with the experiments. Particular attention is paid to the XPM effects in Section 7, where the case of two incident pulses is investigated. Finally, Section 8 gives a brief summary of the main conclusions of the chapter.

2. Propagation Equation As is the case for all electromagnetic phenomena, pulse propagation in optical fibers is governed by Maxwell’s equations. For a nonmagnetic medium, this amounts to solving the wave equation — ¥ — ¥ E = -m0

∂2 D , ∂ t2

(1)

where E is the electric field, m0 is the vacuum permeability, and D is the electric flux density. The constitutive relation between D and E takes a simple form in the frequency domain and can be written as ˜ (r, w ) = e 0 n˜ 2 E ˜ (r, w ), D

(2)

˜ are where e0 is the vacuum permittivity, ñ is the refractive index, and E˜ and D the Fourier transforms of E and D respectively, i.e., ˜ (r, w ) = 1 E 2p

Ú

•

-•

E(r, t) exp(iwt)dt

(3)

˜ (r, w). Using Eqs. (1)–(3), we obtain with a similar relation for D ˜ + (n˜w c) 2 E ˜ = 0, — 2E

(4)

where we have assumed that — · E @ 0. A suitable form for the refractive index in optical fibers is n˜ = n(r, w ) + iac 2w + n2 E , 2

(5)

3. Ultrashort Pulse Propagation in Nonlinear Dispersive Fibers

93

where the linear index n(r, w) is inhomogeneous (in the transverse dimensions x and y) to account for dielectric waveguiding and is frequency dependent to account for chromatic dispersion in silica fibers. The fiber loss is taken into account by the absorption coefficient a. Finally, the last term in Eq. (5) accounts for the fiber nonlinearity. The nonlinear coefficient n2 = 3.2 ¥ 10-16 cm2/W for silica fibers and is nearly frequency independent in the frequency range of interest. It is implicitly assumed that the nonlinear response is instantaneous, an assumption that becomes questionable for ultrashort optical pulses with widths <0.1 ps. Since the refractive index does not vary appreciably along the fiber length (z direction), Eq. (4) can be solved by assuming that ˜ (x, y, z, w ) = ea ˆ ˜ (z, w )y (x, y) = eE ˆ 0y (x, y) exp(ib˜z), E

(6)

where ê is the polarization unit vector, b˜ is the wave number, and y (x, y) is obtained by solving ∂ 2y ∂ 2y Ê n˜ 2w 2 ˜ 2 ˆ + +Á - b ˜ y = 0. ¯ ∂ x 2 ∂y2 Ë c2

(7)

Since the last two terms in Eq. (5) are much smaller than the linear index n(x, y, w), they can be treated by first-order perturbation theory. The complex wave number b˜ is thus well approximated by b˜ @ b (w ) + ia 2 + Db NL , (8) where Db NL =

n 2w c

ÚÚ y (x, y)

2

2

E dxdy

(9)

s

and b(w) is obtained by solving the eigenvalue equation ∂ 2y ∂ 2y Ê w2 ˆ + + Á n(x, y, w ) 2 - b 2 (w )˜ y = 0 2 2 Ë ¯ ∂x ∂y c

(10)

for a given index distribution of the fiber. For a single-mode fiber, y(x, y) is the field distribution of the fundamental fiber mode and has been taken to satisfy the normalization condition

ÚÚ y (x, y)

2

dxdy = 1,

(11)

s

where the integration is carried over the fiber cross section s. The dispersion relation (8) can be used to obtain the envelope propagation equation by noting from Eq. (6) that ∂ a˜ = i [b (w ) + ia 2 + Db NL ]a˜. ∂z

(12)

Since the pulse spectrum is centered at the carrier frequency w0, it is useful to expand b(w) in a Taylor series

94

G.P. Agrawal

b (w ) = b 0 + (w - w 0 )b1 + 12 (w - w 0 ) b 2 + 16 (w - w 0 ) b 3 + . . . , 2

3

(13)

where bn = d nb/dw n is evaluated at w0. The cubic and higher-order terms are generally neglected by assuming that the spectral width Dw << w0. If b2 ⯝ 0 for some specific values of w0 or if the pulses are ultrashort (<0.1 ps), it may be necessary to include the cubic term. It is useful to define the slowly varying pulse amplitude A(z, t) by using a(z, t) = A(z, t) exp[i (b 0 z - w 0t)],

(14)

where a(z, t) is the inverse Fourier transform of ã(z, w). During the Fourier transform operation w - w0 is replaced by the differential operator i(∂/∂t). The use of Eqs. (12)–(14) then leads to the following propagation equation: ∂A ∂A a i ∂ 2A 2 + b1 + A + b 2 2 = ig A A, ∂z ∂t 2 2 ∂t

(15)

where the nonlinearity coefficient g = n2 w 0 cAeff .

(16)

The parameter Aeff is known as the effective fiber core area and from Eq. (9) is given by (noting that E = Ay) -1 Aeff = ÚÚ y (x, y) dxdy. 4

(17)

s

Equation (15) describes the propagation of an optical pulse in single-mode fibers. The pulse envelope moves at the group velocity vg = 1/b1. The group velocity dispersion (GVD) is included through the parameter b2 while the parameter g takes into account the fiber nonlinearity responsible for SPM. Both material dispersion and waveguide dispersion contribute to b2. However, the contribution of material dispersion generally dominates. In that case b2 can be approximated by b 2 ⯝ (l 2pc 2 )D(l ),

(18)

where l = 2pc/w0 is the carrier wavelength and D(l) = l2(d 2n/dl2). For silica fibers D(l) = 0.066 at l = 0.53 mm, and therefore b2 ⯝ 0.06 ps2/m. The nonlinearity parameter g depends on the fiber parameters through Aeff. Its estimated values are in the range 0.01 to 0.02 W-1 m-1. The GVD parameter b2 becomes negative for l > lZD, where lZD is sometimes referred to as the zerodispersion wavelength. In the anomalous dispersion regime (b2 < 0), the fiber can support optical solitons. The wavelength lZD @ 1.3 mm but can be changed in the range 1.2 to 1.6 mm by modifying the fiber design that changes the waveguide dispersion contribution to b2. Depending on the experimental conditions, Eq. (15) may need modification. For example, this equation does not include the effect of SRS that

3. Ultrashort Pulse Propagation in Nonlinear Dispersive Fibers

95

becomes important at high power levels. SRS transfers the pulse energy to the Stokes pulse (shifted downward in frequency by about 13 THz) as the pump pulse propagates inside the fiber. Equation (15) can be generalized to include SRS if we include the Raman gain and consider a coupled set of equations describing the interaction between the pump and Stokes pulses with amplitudes A1 and A2, respectively;

(

∂ A1 1 ∂ A1 1 + + a 1 + g1 A2 ∂ z v g1 ∂ t 2 +

2

)A

1

i ∂ 2 A1 2 b 21 - ig 1 A1 + 2 A2 2 2 ∂t

(

(

2

)A = 0, 1

(19)

)

∂ A2 1 ∂ A2 1 2 + + a 2 - g 2 A1 A2 ∂z vg 2 ∂ t 2 +

i ∂ 2 A2 2 2 b 22 - ig 2 A2 + 2 A1 A2 = 0. 2 2 ∂t

(

)

(20)

Here vgj = b j-1 is the group velocity, gj is the Raman gain coefficient, b2j is the GVD coefficient, and gj is the nonlinear coefficient ( j = 1 and 2 for pump and Stokes pulses, respectively). It is important to include both SPM and XPM in the nonlinear term (Gersten et al., 1980; Schadt and Jaskorzynska, 1987; Agrawal, 1987). Equation (15) should also be modified for ultrashort optical pulses with widths ⱗ0.1 ps. The spectral width Dw of such pulses becomes comparable to the carrier frequency w0, and several approximations made in the derivation of Eq. (15) become questionable. Furthermore, the pulse spectrum is wide enough that the low-frequency components get amplified from the Raman gain at the expense of the high-frequency components. As a result, the spectrum shifts toward shorter frequencies as the pulse propagates inside the fiber, a phenomenon referred to as self-frequency shift (Mitschke and Mollenauer, 1986; Gordon, 1986). One can no longer consider the pump and Stokes pulses separately. Rather than using Eqs. (19) and (20) to describe SRS and related nonlinear effects, a generalization of Eq. (15) is needed. Kodama and Hasegawa (1987) have carried out such a generalization. The resulting propagation equation is ∂A ∂A a i ∂ 2A 1 ∂ 3A 2 + b1 + A = - b 2 2 + ig A A + b 3 3 ∂z ∂t 2 2 ∂t 6 ∂t -a1

(

)

2

∂ ∂A 2 A A - a2 A . ∂t ∂t

(21)

This equation has three additional terms compared with Eq. (15). The term proportional to b3 results from including the cubic term in the expansion (13). The term proportional to a1 is responsible for self-steepening; a1 is generally approximated by a1 ⯝ 2g/w0. The term proportional to a2 includes the effect

96

G.P. Agrawal

of SRS among other things. In particular, the imaginary part of a2 is responsible for the self-frequency shift. The physical origin of this effect is related to the retarded nonlinear response of the fiber. In this chapter we do not consider these effects and only refer to the recent work (Zysset et al., 1987; Bourkoff et al., 1987; Golovchenko et al., 1987; Vysloukh and Matveeva, 1987; Serkin, 1987).

3. Propagation Regimes We first consider the case of pulse propagation below the SRS threshold such that Eq. (15) is applicable. It is useful to define the normalized variables t = (t - b1z) T0 ,

A = P0 exp(-az 2)U ,

(22)

where T0 and P0 are related to the width and the peak power of the incident pulse. Equation (15) then takes the form i

∂U 1 ∂ 2U 1 -az 2 =± e U U, 2 ∂z 2LD ∂t LNL

(23)

where LD = T02 b 2

-1

and LNL = (g P0 ) .

(24)

The choice of the sign in the dispersion term depends on the sign of b2; a plus sign is chosen for normal GVD (b2 > 0) and a minus sign is chosen for anomalous GVD (b2 < 0). The dispersion length LD and the nonlinear length LNL provide the length scales over which dispersive and nonlinear effects become important. Depending on their relative magnitudes, pulse propagation in optical fibers can be classified in three different categories.

3.1 Dispersion-Dominant Regime If the fiber length L is such that L << LNL while L ~ LD, the nonlinear term in Eq. (23) plays a relatively minor role and can be neglected. The resulting linear equation is readily solved to yield •

U (z, t ) = Ú U˜ (0, w ) exp[-iwt ± iw 2 (z 2LD )]dw , -•

(25)

where U˜ (0, w) is the Fourier transform of the incident pulse amplitude U(0, t). Equation (25) has been widely used to study pulse broadening in the dispersion-dominant regime. In particular, a simple analytic expression for the pulse width can be obtained for Gaussian pulses (Marcuse, 1981). For an unchirped Gaussian pulse, U(0, t) = exp(-t 2/2), and the pulse broadens by a factor of (1 + z2/L2D)1/2 as it propagates along the fiber. The dispersion length LD corresponds to the fiber length over which the pulse width increases by

3. Ultrashort Pulse Propagation in Nonlinear Dispersive Fibers

97

about 40% (in the absence of nonlinear effects). The dispersion-dominant regime or Eq. (25) is applicable when the fiber and pulse parameters are such that LD g P0T02 = << 1. LNL b2

(26)

As a rough estimate, P0 should be less than 10 mW for 10-ps pulses.

3.2 Nonlinearity-Dominant Regime If the fiber length L is such that L << LD while L ⲏ LNL, the dispersion term in Eq. (23) plays a relatively minor role and can be neglected as long as the pulse shape is relatively smooth so that |∂ 2U/∂t 2| is not too large. The resulting equation can be solved analytically with the result U (z, t ) = U (0, t ) exp[if SPM (z, t )],

(27)

Ê 1 - e -az ˆ 2 f SPM (z, t ) = Á ˜ U (0, t ) . Ë aLNL ¯

(28)

where

Equation (27) shows that the nonlinearity gives rise to SPM while it does not affect the pulse shape. The effect of SPM is to broaden the pulse spectrum with considerable internal structure. Figure 3.1 shows the calculated SPM spectra for a Gaussian pulse obtained by taking the Fourier transform of Eq. (27) with a = 0 and z = 40LNL. Quite generally, the number of internal peaks increases linearly with the fiber length (for a = 0), and the dominant peaks occur near the spectral boundaries. The spectral range is determined by Dwmax = |∂fSPM/∂t|max, which is the maximum chirp induced by the nonlinearity. The SPM-induced spectral features have been observed in optical fibers and were used to estimate n2 (Stolen and Lin, 1978).

3.3 Dispersive Nonlinear Regime When the fiber length L is longer than or comparable to both LD and LNL, dispersion and nonlinearity act together as the pulse propagates along the fiber. Their mutual interaction plays an important role and has been used to generate solitons and to compress optical pulses. To understand this interaction theoretically, Eq. (23) is solved numerically and has been found to be extremely helpful in interpreting the experimental results (Mollenauer et al., 1980; Grischowsky and Balant, 1982; Tomlinson et al., 1984). It is useful to normalize the fiber length by introducing x = z/LD and to write Eq. (23) in the following normalized form: i

∂U 1 ∂ 2U 2 =± - N 2e -az U U , ∂x 2 ∂t2

(29)

98

G.P. Agrawal

Figure 3.1. Calculated SPM spectrum of a Gaussian pulse (in the absence of dispersion) at a distance z = 40LNL, where LNL is the nonlinear length defined by Eq. (24).

where N2 =

LD gP0T02 = . LNL b2

(30)

In the lossless case (a = 0), Eq. (29) is often referred to as the nonlinear Schrödinger equation (NSE). In the anomalous dispersion regime (corresponding to the choice of minus sign), the NSE is known to have exact solutions, known as solitons (Zakharov and Shabat, 1972; Hasegawa and Tappert, 1973). For an initial pulse shape U(0, t) = sech(t) and integer values of the parameter N, the solitons follow a periodic evolution pattern with the period x0 = p/2. In the unnormalized units, the soliton period z0 becomes z0 =

p pT02 LD = . 2 2 b2

(31)

The fundamental soliton corresponds to N = 1 and propagates without change in its shape. From Eq. (30), the peak power necessary to excite the fundamental soliton is P1 = |b2|/gT20. For a hyperbolic secant pulse, the pulse width Tp (FWHM) is related to T0 by Tp ⯝ 1.76T0. This relation should be

3. Ultrashort Pulse Propagation in Nonlinear Dispersive Fibers

99

used for comparison with experiments. As a rough estimate, for 1.55-mm solitons in silica fibers, z0 ~ 25 m and P1 ~ 1 W when Tp = 1 ps. In the next two sections we consider the pulse propagation characteristics in the normal and anomalous GVD regimes based on the numerical solution of Eq. (29). A fast Fourier transform (FFT)-based beam propagation method (Fleck et al., 1976; Lax et al., 1981; Agrawal and Potasek, 1986) was used for the numerical solution. Although a finite-difference scheme can be used, the FFT method is considerably faster for a given accuracy (typically by more than a factor of 10). In the FFT method, Eq. (29) is formally written as ∂U = (D + Q)U , ∂x

(32)

where D=±

i ∂2 2 ∂t 2

2

and Q = iN 2e -az U .

(33)

The field is propagated inside the fiber by a small distance d using the following prescription:

(Ú

È Ê dD ˆ U (x + d , t ) = Íexp exp Î Ë 2 ¯

x +d

x

)

Ê dD ˆ ˘ Q(x ¢)dx ¢ exp U (x, t ). Ë 2 ¯ ˙˚

(34)

The numerical procedure consists of propagating the field by a distance d/2 with dispersion only, multiplying the result by a nonlinear term that represents the effect of nonlinearity over the whole step length d, and then propagating the field for the remaining distance d/2 with dispersion only. In effect, the nonlinearity is assumed to be lumped at the midplane of each segment. This technique is a generalization of the split-step method (Hasegawa and Tappert, 1973; Fisher and Bischel, 1975). The propagation in a linear dispersive medium indicated by the exponential operator exp(dD/2) in Eq. (34) can be accomplished using the Fourier transform method. The use of the FFT algorithm makes this step relatively fast. The integral in Eq. (34) is well approximated using the trapezoidal rule:

Ú

x +d

x

1 Q(x ¢)dx ¢ ⯝ d [Q(x ) + Q(x + d )]. 2

(35)

However, Q(x + d) cannot be evaluated since U(x + d, t) is not known while evaluating Eq. (35) at the midsegment located at x + d/2. One can approximate Q(x + d ) by Q(x) if |U|2 has not significantly changed in going from x to x + d. This approximation can, however, restrict the step size d to relatively small values. Another approach is to follow an iterative procedure in which Eq. (35) is evaluated several times with increasingly accurate values of Q(x + d ). In practice, two iterations are often sufficient to obtain the desired accuracy.

100

G.P. Agrawal

4. Normal Dispersion In the normal dispersion regime, the GVD parameter b2 > 0. For silica fibers, this regime corresponds to l < 1.3 mm and has been extensively studied because of the availability of solid-state and dye lasers in the visible region. At l = 0.53 mm, b2 ⯝ 0.06 ps2m and g ⯝ 0.01 W-1 m-1 for typical fiber parameters. From Eq. (24), LD ~ 100 m for T0 ⯝ 3–5 ps and LNL ~ 1 m for P0 ~ 100 W. Thus the parameter N ~ 10 under typical experimental conditions, although N ~ 100 is feasible for wider and more intense pulses. Figure 3.2 shows the evolution of pulse shape along the fiber for N = 30 after assuming a = 0 and a Gaussian pulse with U(0, t) = exp(-t 2/2). As the pulse propagates, it broadens and develops a nearly rectangular profile with sharp leading and trailing edges. The combination of rapidly varying intensity and SPM in these steep-slope regions broadens the pulse spectrum. Because the new frequency components are mainly generated near the edges, the pulse develops a nearly linear frequency chirp across its entire width (Grischkowsky and Balant, 1982; Tomlinson et al., 1984). This linear chirp helps to compress the pulse by passing it subsequently through a dispersive delay line (often a grating pair). An interesting feature of Figure 3.2 is the presence of rapid oscillations in the wings of the pulse. Tomlinson et al. (1985) have interpreted these oscillations in terms of optical wave breaking resulting from a mixing of the SPMinduced frequency-shifted components with the unshifted light in the wings. This phenomenon can also be understood as a four-wave mixing process and should be manifest through the presence of sidelobes in the pulse spectrum. Figure 3.3 shows the pulse shape and spectrum at x = 0.08. As seen there, oscillations near the pulse edges are accompanied by the sidelobes in the pulse spectrum. The central structure in the spectrum is due to SPM (see Figure 3.1).

Figure 3.2. Evolution of the pulse shape of an unchirped Gaussian pulse along the fiber length (x = z/LD) for N = 30 in the normal dispersion regime. The peak power is related to N through Eq. (30).

3. Ultrashort Pulse Propagation in Nonlinear Dispersive Fibers

101

Figure 3.3. Pulse shape and spectrum of an unchirped Gaussian pulse at x = 0.08 for N = 30. Sidelobes in the spectrum and oscillations near the pulse edges are due to optical wave breaking.

The results shown in Figures 3.2 and 3.3 are for an initially unchirped pulse. The pulses emitted by practical laser sources are often chirped and may follow quite a different evolution pattern than that shown in Figure 3.2 (Lassen et al., 1985). To include the effect of initial chirp, the incident amplitude is taken to be U (0, t ) = exp[- 12 (1 + iC )t 2 ],

(36)

where C is a measure of chirp. C can be estimated by noting that the spectral width of the chirped pulse is larger by a factor of (1 + C2)1/2 than the transform-limited value for C = 0. Figure 3.4 shows the pulse shape and spectrum under conditions identical to those of Figure 3.3 except for the chirp parameter that has a value of C = 20. A comparison of Figures 3.3 and 3.4 shows how much an initial chirp can modify the propagation behavior. For example, the shape for the initially chirped pulse is nearly triangular rather than rectangular. The pulse evolution is also sensitive to the fiber loss (Lassen et al., 1985). Thus, for an actual comparison between theory and experiment it is necessary to include both the chirp and the loss in numerical simulations.

5. Anomalous Dispersion In the anomalous dispersion regime, the GVD parameter b2 < 0. For silica fibers, this regime corresponds to l > 1.3 mm and has been extensively studied, using color-center lasers, for example (Mollenauer et al., 1980). At l = 1.55 mm, b2 ⯝ -0.02 ps2/m and g ⯝ 0.01 W-1 m-1 for typical fiber parameters. For T0 ⯝ 1 ps, the dispersion length LD ⯝ 50 m; the peak power correspond-

102

G.P. Agrawal

Figure 3.4. Pulse shape and spectrum for parameters identical to those of Figure 3.3 except for the chirp parameter, which has a value C = 20 instead of 0.

ing to the fundamental soliton (N = 1) can be estimated from Eq. (30) and is P0 ~ 1 W. In the absence of fiber loss, the fundamental soliton can propagate undistorted for arbitrarily long distances. This is the reason behind the current interest in soliton-based communication systems (Hasegawa and Tappert, 1973). In practice, however, the fiber loss leads to a gradual broadening of the soliton. Periodic compensation of the loss by Raman gain has been proposed to restore the original soliton (Mollenauer et al., 1986). The higher-order solitons (N > 1) follow a periodic evolution pattern along the fiber with a period z0 = (p/2)LD. Figure 3.5 shows, as an example, the evolution pattern of an N = 3 soliton over one period obtained by solving Eq. (29) with N = 3, a = 0, and U(0, t) = sech(t). The pulse initially narrows, develops a two-peak structure, and then reverses its propagation behavior beyond z/z0 = 0.5 such that the original pulse is restored at z = z0. Both pulse narrowing and pulse restoration at the soliton period have been observed experimentally (Stolen et al., 1983). In particular, initial narrowing of the higher-order soliton has been used to compress the optical pulses by suitably selecting the peak power and the fiber length. A combination of a normal dispersion fiber with a grating pair followed by an anomalous dispersion fiber has been used to compress optical pulses by more than three orders of magnitude (Tai and Tomita, 1986) and to generate pulses as short as 33 fs (Gouveia-Neto et al., 1987). The soliton formation capacity of optical fibers has led to the development of the soliton laser (Mollenauer and Stolen, 1984). A piece of single-mode fiber inside the cavity is used to shape the pulses. The pulse width can be controlled by adjusting the fiber length. Pulses as short as 50 fs have been generated directly from a soliton laser and compressed to about 19 fs using

3. Ultrashort Pulse Propagation in Nonlinear Dispersive Fibers

103

Figure 3.5. Evolution of N = 3 soliton over one period. Note the initial pulse narrowing in the anomalous dispersion regime.

the high-order soliton effect in an external fiber (Mitschke and Mollenauer, 1987). Equation (29) has proved to be very useful in understanding the propagation behavior in the anomalous dispersion regime of optical fibers. However, as discussed in Section 2, for ultrashort pulses (T0 ⱗ 0.1 ps) Eq. (29) is no longer adequate, and Eq. (21) should be used in its place. The necessity of including the higher-order nonlinear terms in Eq. (21) was established in a recent experiment where a new effect known as the self-frequency shift was observed (Mitschke and Mollenauer, 1986). In this phenomenon the center frequency of the soliton shifts toward red as the pulse propagates inside the fiber. Physically, the low-frequency components are amplified from the Raman gain at the expense of the high-frequency components. Equation (21) includes the Raman gain through the last term and can be used to study such self-frequency shifts (Gordon, 1986; Kodama and Hasegawa, 1987). The effect of higher-order dispersion on the soliton behavior [governed by the term proportional to b3 in Eq. (21)] has also attracted attention. In general, such effects become significant only when the pulse wavelength nearly coincides with the zero-dispersion wavelength of the fiber and can lead to considerable pulse distortion (Agrawal and Potasek, 1986). It has been shown that the fiber can support a soliton even at the zero-dispersion wavelength (Wai et al., 1987) by shifting the center wavelength toward the anomalous dispersion regime.

104

G.P. Agrawal

6. Stimulated Raman Scattering In preceding sections the peak power of the incident pulse has been assumed to be considerably below the SRS threshold (Smith, 1972). When this condition is not satisfied, the Raman gain provided by the fiber leads to a rapid buildup of the Stokes pulse. Figure 3.6 shows the Raman gain spectrum for silica fibers (Stolen, 1980). The large spectral width results from the amorphous nature of fused silica. Since the largest gain occurs at a frequency shifted downward by about 13 THz from the pump frequency, the Stokes pulse appears red-shifted by that amount. If the incident pulse is so short (ⱗ0.1 ps) that its spectral width exceeds the Raman shift, the distinction between the pump and Stokes pulses is not sharp since their spectra overlap. In that case pulse evolution is governed by Eq. (21). However, for pulse widths ⲏ1 ps, the pump and Stokes pulses can be treated separately, and their propagation in optical fibers is described by the coupled set of Eqs. (19) and (20). In this section we consider numerical solution of these equations and compare the results with the experimental work on SRS. The numerical method is a generalization of that used to solve the NSE, described in Section 3.3.

Figure 3.6. Raman gain spectrum for silica fibers (after Stolen, 1980) The peak gain is about 1 ¥ 10-13 m/W for a 1-mm pump and scales as l p-1 with change in the wavelength.

3. Ultrashort Pulse Propagation in Nonlinear Dispersive Fibers

105

For the purpose of computations, it is convenient to define the reduced time T = t - z v g1

(37)

and write Eqs. (19) and (20) in the form

(

∂ Ap 1 + a p + g p As ∂z 2

(

2

)A

∂ As ∂ As 1 +d + a s - g s Ap ∂z ∂T 2

p

2

+

∂ 2 Ap i 2 bp - ig p Ap + 2 As 2 2 ∂T

(

)A

s

+

2

)A

i ∂ 2 As 2 bs - ig s As + 2 Ap 2 ∂T 2

(

s

2

= 0,

)A

p

(38)

= 0, (39)

where we have used the subscripts p and s to identify the pump and Stokes variables. The parameter d=

v gp - v gs v gpv gs

(40)

accounts for the group velocity mismatch between the pump and Raman pulses. Because of the Stokes shift of about 13 THz, the parameters bj, gj, and gj are slightly different for j = p and s. In particular, form Eqs. (16) and (18) b s = (l s l p )b p ,

g s = (l p l s )g p ,

(41)

where lp and ls are the center wavelengths for the pump and Raman pulses, respectively. Similarly, the gains are related by g s = (l p l s )g p .

(42)

For definiteness, the numerical results for SRS are presented here for lp ⯝ 0.53 mm; qualitatively similar behavior is expected to occur for other wavelengths. Using typical values for the fiber parameters, we estimate that ap = 0.0012 m-1, bp = 0.06 ps2m-1, gp = 0.015 W-1 m-1, and gp = 0.015 W-1 m-1. Assuming a Gaussian unchirped pump pulse, the incident field Ap(0, t) is taken to be Ap (0, t ) = P0 exp(-T 2 2T 02 ).

(43)

We choose T0 = 10 ps, which corresponds to a full width at half-maximum (FWHM) of about 16 ps. The peak pump power is taken to be P0 = 500 W. Numerical simulations require the use of a seed Raman pulse. It was taken to be a Gaussian pulse with a width identical to that of the pump pulse but with a power of 10-6 of the pump pulse. The two pulses overlap initially but the Raman pulse travels faster than the pump pulse inside the fiber because of the group velocity mismatch. The walk-off effect is included through the parameter d in Eq. (39); its value is estimated to be about 5 ps/m. Because of the use of a specific seed pulse, the results presented here are applicable more directly to the case of Raman amplification rather than noise-induced SRS. Most of the qualitative features are, however, expected to apply to SRS as well. Figure 3.7 shows the evolution of the pump and Raman pulses as they copropagate inside a 5-m-long fiber. As the Raman pulse moves ahead of the

106

G.P. Agrawal

Figure 3.7. Evolution of pump pulse and Raman pulse along the fiber length. Pump pulse parameters are lp = 0.53 mm, T0 = 10 ps, and P0 = 500 W. Pulse walk-off is included by taking d = -5 ps/m.

pump pulse, the energy for the generation of the Raman pulse comes from the leading edge of the pump pulse. This is seen clearly in Figure 3.8, where the pulse shapes are shown overlapped at z = 2 m and 5 m. At z = 2 m, the creation of the Raman pulse has led to a two-peak structure for the pump pulse as a result of pump depletion. The peak near the leading edge disappears with further propagation as the Raman pulse walks through it. At z = 5 m, the two pulses are completely separated. Because of the walk-off effect, both pulses are asymmetric and narrower than the incident pulse. In particular, the pump pulse consists of the trailing portion of the incident pulse that remained undepleted because of the walk-off of the Raman pulse. The spectral evolution of the two pulses along the fiber length is shown in Figure 3.9. Both pulses exhibit asymmetric spectral broadening with

3. Ultrashort Pulse Propagation in Nonlinear Dispersive Fibers

107

Figure 3.8. Comparison of pump and Raman pulse shapes at z = 2 m and z = 5 m with parameters of Figure 3.7. At z = 2 m, the pump pulse develops a two-peak structure because of pump depletion occurring at the location of the Raman pulse. Two pulses are completely separated after propagating a distance of 5 m.

Figure 3.9. Evolution of pump and Raman pulse spectra along the fiber length. The parameter values are identical to those of Figure 3.7.

108

G.P. Agrawal

Figure 3.10. Comparison of pump and Raman pulse spectra at z = 5 m. Note the spectral broadening of the Raman pulse by more than a factor of 2 compared with that of the pump pulse.

Figure 3.11. Experimental spectra of pump and Raman pulses after a 25-ps pump pulse at 532 nm has propagated 10 m inside a fiber. Four spectra correspond to different pump pulse energies normalized to E0, an energy below the Raman threshold. (After Alfano et al., 1987.)

3. Ultrashort Pulse Propagation in Nonlinear Dispersive Fibers

109

considerable internal structure. In general, the spectral width of the Raman pulse exceeds that of the pump pulse by a factor of 2 or more. These features are more clearly seen in Figure 3.10, where the spectra at z = 5 m are shown on an expanded scale. Note that the spectral range of the Raman pulse is about 2.5 times that of the pump pulse. This feature was predicted by Gersten et al. (1980) and has been verified by Alfano et al. (1987). Figure 3.11 shows their experimentally observed spectra at the output end of a 10-m-long fiber for a 25-ps pump pulse with increasing pump pulse energies. The Raman line is broader by about a factor of 2.8. The spectral asymmetry of the pump line is also evident. Similar qualitative features have been observed at other pump wavelengths. Figure 3.12 shows, as an example, the observed pump spectra at various peak power levels when 140-ps-wide pump pulses at 1.06 mm are propagated along a 150-m-long fiber (Kean et al., 1987). For peak powers below 100 W, the observed spectra are due to SPM (see Figure 3.1). The Raman threshold is reached near 100 W. The spectrum at 148 W shows the asymmetric spectral broadening resulting from the Raman interaction and should be compared with that shown in Figure 3.10. The physical origin of

Figure 3.12. Experimental spectra of the pump pulse at different peak powers after a 140-ps pulse at 1.06 mm has propagated inside a 150-m-long fiber. The Raman threshold is reached at about 100 W. (After Dean et al., 1987.)

110

G.P. Agrawal

spectral broadening is related to XPM, while the physical origin of spectral asymmetry is related to the pulse walk-off. These features are discussed further in the following section.

7. Cross-Phase Modulation Although the major qualitative features seen in the pulse spectra of Figures 3.9 and 3.10 are due to XPM, it is difficult to isolate the XPM-induced features. For this reason, we consider in this section the interaction of two incident pulses copropagating along the fiber without SRS; that is, gp = 0 and gs = 0 in Eqs. (38) and (39). The other parameters are identical to those used in Section 6 except for the incident peak powers, which we take to be P1 = P2 = 200 W. For these power levels both pulses are below the Raman threshold for L = 5 m; that is, gsPL < 16 (Smith, 1972; Stolen, 1980). Both incident pulses are assumed to be Gaussian with T0 = 10 ps; they overlap completely at the input end of the fiber. The pulse walk-off effects are included by taking d = -2 ps/m, a value appropriate when the center wavelengths of two pulses are spaced by about 5 nm. Similar to the SRS case shown in Figures 3.7 and 3.9, the evolution of the pulse shapes and spectra along a 5-m-long fiber was studied. It was found that the shapes of both pulses remain largely unaffected during XPM interaction except for the walk-off effect; that is, the peaks of the Gaussian pulses drifted apart at a rate of about 2 ps/m, as one would have expected. The spectral evolution is, however, governed by the combined effect of SPM and XPM. Figure 3.13 shows the pulse spectra at z = 5 m for the two pulses. Similar to the case of SPM (see Figure 3.1), the spectra exhibit inter-

Figure 3.13. Calculated spectra of two pulses at z = 5 m. Both pulses are identical (except for their center wavelengths) and launched simultaneously inside the fiber with parameters T1 = T2 = 10 ps and P1 = P2 = 200 W. Group velocity mismatch is included by taking d = -2 ps/m.

3. Ultrashort Pulse Propagation in Nonlinear Dispersive Fibers

111

Figure 3.14. Same as in Figure 3.13 except that the group velocity mismatch is ignored by setting d = 0. Since the spectra are identical, only the spectrum of one pulse is shown. The right and left plots show the spectrum without and with XPM, respectively. Note the XPM-induced broadening by a factor of about 3.

nal structure with multiple peaks. The noticeable new features are (1) that the spectrum of each pulse is asymmetric and (2) that the spectrum of one pulse is the mirror image of the other (shifted by the initial wavelength separation of the two pulses). In general, blue-shifted components are more intense for the slower pulse while the opposite occurs for the fast-moving pulse. In order to isolate the features related to XPM and group velocity msmatch, Figure 3.14 shows the spectrum for the case d = 0. Also included for comparison is the spectrum expected in the absence of XPM. Since the spectra of both pulses are found to be identical for d = 0, the spectra of only one pulse are shown in Figure 3.15. A comparison of the two spectra in Figure 3.14 shows that in the absence of group velocity mismatch (d = 0), the effect of XPM is to broaden the spectrum by nearly a factor of 3 over that expected by SPM alone. By comparing Figures 3.13 and 3.14 we conclude that it is the pulse walk-off that is responsible for the spectral asymmetry seen in Figure 3.13. It should, however, be stressed that XPM is also necessary for spectral asymmetry since it is XPM that provides the physical mechanism for two pulses to interact with each other. A simple analytic model for the XPM interaction between two copropagating pulses can be developed to understand the origin of spectral features seen in Figures 3.13 and 3.14. The model makes the following three assumptions to simplify the coupled-amplitude Eqs. (19) and (20). (1) The effects of GVD are negligible so that b21 = b22 = 0. This approximation is justified if the fiber length is much shorter than the dispersion length LD associated with each pulse [see Eq. (23)]. (2) The peak powers of the incident pulses are below the Raman threshold so that g1 = g2 = 0. (3) The fiber loss is negligible so that a1 = a2 = 0. The last assumption is not necessary but is made to simplify the

112

G.P. Agrawal

treatment. The inclusion of fiber loss does not affect the conclusions drawn here. It can readily be verified by using Eqs. (19) and (20) with these simplifications that the pulse shapes remain unchanged during propagation. The effect of SPM and XPM is to generate a nonlinear phase shift accumulated over the fiber length L (Islam et al., 1987). The phase shift for pulse 1 is given by F1 (T ) = g 1 Ú

L

0

[ A (0, T ) 1

2

]

+ 2 A2 (0, T - zd ) dz, 2

(44)

where T is the reduced time given by Eq. (37) and d is the group velocity mismatch defined in a way similar to Eq. (40). If the two pulses are Gaussian in shape and are launched simultaneously with complete overlap, the incident fields are given by Am (0, T ) = Pm exp(-T 2 2Tm2 ),

(45)

where m = 1 or 2, Pm is the peak power, and Tm is the 1/e half-width. We assume equal pulse widths, T1 = T2 = T0, but allow for unequal peak powers. By using Eq. (45) in Eq. (44) and performing the integration, we obtain p ¸ Ï F1 (t ) = g 1L ÌP1 exp(-t 2 ) + P2 [erf (t ) - erf (t - d )]˝, d ˛ Ó

(46)

where erf stands for the error function, t = T/T0 is the normalized time coordinate, and d = Ld T0

(47)

is the normalized walk-off parameter. The nonlinearity-induced chirp is obtained by using D␯1 = -(1/2p)∂F1/∂T. By differentiating Eq. (46), it is given by D␯1 (t ) =

g 1L - t 2 Ï P2 ¸ e ÌP1t + [exp(2td - d 2 ) - 1]˝. pT0 d ˛ Ó

(48)

To the lowest order in d, the chirp is given by the simple relation D␯1 (t ) ⯝

g 1L - t 2 e [P1t + P2 (2t - d )]. pT0

(49)

The chirp D␯2 can be obtained by interchanging the subscripts 1 and 2 and replacing d by -d in Eq. (49) and is given by D␯ 2 (t ) =

g 2L -t2 e [P2t + P1 (2t + d )]. pT0

(50)

Equations (49) and (50) can be used to understand all the spectral features seen in Figures 3.13 and 3.14. Consider first the case d = 0 (no walk-off), for which the numerical pulse spectra are shown in Figure 3.14. Since P1 = P2 was assumed in numerical simulations, Eq. (49) indicates that the effect of

3. Ultrashort Pulse Propagation in Nonlinear Dispersive Fibers

113

XPM is to broaden the pulse spectrum by a factor of 3 over that expected from the SPM alone. This is in agreement with the results of Figure 3.14. The fact that the broadening factor is slightly smaller than 3 in Figure 3.14 is related to the GVD, which reduces the peak powers slightly by broadening the pulse during its propagation inside the fiber. In the presence of group velocity mismatch (d π 0), Eq. (49) shows that the chirp can be larger or smaller depending on the relative signs of t and d. For the results shown in Figure 3.13, d < 0. As a consequence, the chirp is larger near the trailing edge (t > 0) and smaller near the leading edge (t < 0) for pulse 1 while the opposite occurs for pulse 2. Since blue components are generated near the trailing edge, they are expected to be more intense for pulse 1. For the same reason, the spectral spread should be larger toward the blue side than the red side for pulse 1. This feature is clearly seen in Figure 3.13. Using a similar argument, Eq. (50) predicts that the spectrum of pulse 2 will have a larger spread on the red side, and the red components will be more intense. For the specific case of gi ⯝ g2 and P1 = P2, the two spectra should be mirror images of each other since D␯2(t) = -D␯1(-t). This feature would, of course, be absent for unequal peak powers. Equations (49) and (50) can be used to predict the relative spectral widths of two spectra for the case of P1 π P2. Since the spectral range is relatively unaffected by pulse walk-off (compare Figures 3.13 and 3.14), we set d = 0. The ratio of the spectral widths is then given by D␯ 2 l1 Ê P2 + 2P1 ˆ = , D␯1 l 2 Ë P1 + 2P2 ¯

(51)

where we used Eq. (16) to show the dependence on wavelengths. For l1 ⯝ l2, the width ratio can vary from 0.5 to 2 depending on the power ratio P2/P1. In particular, when a weak probe pulse copropagates with an intense pulse (P2 << P1), its spectrum is broadened by a factor of more than 2 over that of the pump pulse because of XPM (Gersten et al., 1980; Alfano et al., 1987). Note, however, that the width ratio can be quite different if l1 and l2 differ significantly. For example, the spectral broadening would be by a factor of 4 if the probe pulse is at the second harmonic of the pump pulse.

8. Conclusion This chapter has reviewed the propagation characteristics of ultrashort optical pulses in single-mode fibers influenced by the dispersive and nonlinear effects. When the pulse peak power is below the Raman threshold, the propagation behavior is modeled well by the nonlinear Schrödinger equation (29). New qualitative features arise depending on whether the propagation occurs in the normal or the anomalous dispersion regime. In the latter case, the fiber supports optical solitons and has found applications in the design of the soliton laser. When the pulse peak power exceeds the Raman thresh-

114

G.P. Agrawal

old, the evolution of the pump and Raman pulses is governed by the coupled set of Eqs. (38) and (39). In this case it is important to take into account both the group velocity mismatch and XPM. In particular, XPM is responsible for new spectral features such as an additional broadening of the Raman pulse spectrum by a factor of 2 or more over that of the pump pulse. Of course, SRS is not a prerequisite for the observation of these novel qualitative features; similar propagation characteristics can be observed by launching simultaneously two optical pulses inside the fiber. Clearly, optical fibers are extremely versatile for studying nonlinear phenomena and should find new applications, including one in the design of ultrafast supercontinuum laser sources.

References Agrawal, G.P. (1987) Modulation instability induced by cross-phase modulation. Phys. Rev. Lett. 59, 880–883. Agrawal, G.P. and M.J. Potasek (1986) Nonlinear pulse distortion in single-mode optical fibers at the zero-dispersion wavelength. Phys. Rev. A 33, 1765–1776. Alfano, R.R. (1985) Ultrafast supercontinuum laser source. Proc. Lasers ’85, pp. 110– 122. Alfano, R.R., P.L. Baldeck, F. Raccah, and P.P. Ho (1987) Cross-phase modulation measured in optical fibers. Appl. Opt. 26, 3491–3492. Alfano R.R. and S.L. Shapiro (1970) Observation of self-phase modulation in crystals and glasses. Phys. Rev. Lett. 24, 592–594. Baldeck, P.L., P.P. Ho, and R.R. Alfano (1987) Effects of self, induced and cross phase modulations on the generation of picosecond and femtosecond white light supercontinua. Rev. Phys. Appl. 22, 1677–1694. Basch, E.E., ed. (1986) Optical-Fiber Transmission. Sams, Indianapolis, Indiana. Bourkoff, E., W. Zhao, R.L. Joseph, and D.N. Christodoulides (1987) Evolution of femtosecond pulses in single-mode fibers having higher-order nonlinearity and dispersion. Opt. Lett. 12, 272–274. Fisher, R.A. and W.K. Bischel (1975) Numerical studies of the interplay between selfphase modulation and dispersion for intense plane-wave laser pulses. J. Appl. Phys. 46, 4921–4934. Fleck, J.A., J.R. Morris, and M.D. Feit (1976) Time-dependent propagation of highenergy laser beams through the atmosphere. Appl. Phys. 10, 129–160. Fork, R.L., C.H. Brito Cruz, P.C. Becker, and C.V. Shank (1987) Compression of optical pulses to six femtoseconds by using cubic phase compensation. Opt. Lett. 12, 483–485. Gersten, J.I., R.R. Alfano, and M. Belic (1980) Combined stimulated Raman scattering and continuum self-phase modulations. Phys. Rev. A 21, 1222–1224. Golovchenko, E.A., E.M. Dianov, A.N. Pilipetskii A.M. Prokhorov, and V.N. Serkin (1987) Self-effect and maximum contraction of optical femtosecond wave packets in a nonlinear dispersive medium. JETP Lett. 45, 91–95. Gordon, J.P. (1986) Theory of the soliton self-frequency shift. Opt. Lett. 11, 662–664. Gouveia-Neto, A.S., A.S.L. Gomes, and J.R. Taylor (1987) Generation of 33-fsec pulses at 1.32 mm through a high-order soliton effect in a single-mode optical fiber. Opt. Lett. 12, 395–397.

3. Ultrashort Pulse Propagation in Nonlinear Dispersive Fibers

115

Grischkowsky, D. and A.C. Balant (1982) Optical pulse compression based on enhanced frequency chirping. Appl. Phys. Lett. 41, 1–3. Hasegawa, A. (1983) Amplification and reshaping of optical solitons in glass fiber— IV. Opt. Lett. 8, 650–652. Hasegawa, A. and F. Tappert (1973) transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion. App. Phys. Lett. 23, 142–144. Islam, M.N., L.F. Mollenauer, R.H. Stolen, J.R. Simpson, and H.T. Shang (1987) Cross-phase modulation in optical fibers. Opt. Lett. 12, 625–627. Kean, P.N., K. Smith, and W. Sibbett (1987) Spectral and temporal investigation of self-phase modulation and stimulated Raman scattering in a single-mode optical fiber. IEE Proc. 134, 163–170. Kodama, Y. and A. Hasegawa (1987). Nonlinear pulse propagation in a monomode dielectric guide. IEEE J. Quantum Electron. QE-23, 510–524. Lassen, H.E., F. Mengel, B. Tromborg, N.C. Albertsen, and P.L. Christiansen (1985). Evolution of chirped pulses in nonlinear single-mode fibers. Opt. Lett. 10, 34–36. Lax, M., J.H. Batteh, and G.P. Agrawal (1981) Chenneling of intense electromagnetic beams. J. Appl. Phys. 52, 109–125. Marcuse, D. (1981) Pulse distortion in single-mode fibers. Appl. Opt. 19, 1653–1660. Miller, S.E. and I.P. Kaminow, eds. (1988) Optical Fiber Telecommunications II. Academic Press, Boston, Massachusetts. Mitschke, F.M. and L.F. Mollenauer (1986) Discovery of the soliton self-frequency shift. Opt. Lett. 11, 659–661. Mitschke, F.M. and L.F. Mollenauer (1987) Ultrashort pulses from the soliton laser. Opt. Lett. 12, 407–409. Mollenauer, L.F. and R.H. Stolen (1984) The soliton laser. Opt. Lett. 9, 13–15. Mollenauer, L.F., R.H. Stolen, and J.P. Gordon (1980) Experimental observation of picosecond pulse narrowing and solitons in optical fibers. Phys. Rev. Lett. 45, 1095–1097. Mollenauer, L.F., J.P. Gordon, and M.N. Islam (1986) Soliton propagation in long fibers with periodically compensated loss. IEEE J. Quantum Electron. QE-22, 157–173. Nikolaus, B. and D. Grischkowsky (1983) 12 ¥ pulse compression using optical fibers. Appl. Phys. Lett. 42, 1–2. Schadt, D. and B. Jaskorzynska (1987) Frequency chirp and stimulated Raman scattering influenced by pulse walk-off in optical fibers. J. Opt. Soc. Am. 4, 856–862. Serkin, V.N. (1987) Colored envelope solitons in optical fibers. Sov. Tech. Phys. Lett. 13, 320–321. Shank, C.V., R.L. Fork, R. Yen, R.H. Stolen, and W.J. Tomlinson (1982) Compression of femtosecond optical pulses. Appl. Phys. Lett. 40, 761–763. Shank, C.V., R.L. Fork, C.H. Brito Cruz, and W. Knox (1986). In Ultrafast Phenomena V, G.R. Fleming and A.E. Siegman, eds. Springer-Verlag, Heidelberg. Smith, R.G. (1972) Optical power handling capacity of low loss optical fibers as determined by stimulated Raman and Brillouin scattering. Appl. Opt. 11, 2489–2494. Stolen, R.H.(1980) Nonlinearity in fiber transmission. Proc. IEEE 68, 1232–1236. Stolen, R.H. and C. Lin (1978) Self-phase modulation in silica optical fibers. Phys. Rev. A 17, 1448–1453. Stolen, R.H., L.F. Mollenauer, and W.J. Tomlinson (1983) Observation of pulse restoration at the soliton period in optical fibers. Opt. Lett. 8, 186–188.

116

G.P. Agrawal

Tai, K. and A. Tomita (1986) 1100 ¥ optical fiber pulse compression using grating pair and soliton effect at 1.319 mm. Appl. Phys. Lett. 48, 1033–1035. Tomlinson, W.J., R.H. Stolen, and C.V. Shank (1984) Compression of optical pulses chirped by self-phase modulation in fibers. J. opt. Soc. Am. B 1, 139–149. Tomlinson, W.J., R.H., Stolen, and A.M. Johnson (1985) Optical wave breaking in nonlinear optical fibers. Opt. Lett. 10, 457–459. Vysloukh, V.A. and T.A. Matveeva (1987) Influence of inertia of nonlinear response on compression of femtosecond pulses. Sov. J. Quantum Electron. 17, 498–500. Wai, P.K., C.R. Menyuk, H.H. Chen, and Y.C. Lee (1987) Soliton at the zero-groupdispersion wavelength of a single-mode fiber. Opt. Lett. 12, 628–630. Zakharov, V.E. and A.B. Shabat (1972) Exact theory of two-dimensional selffocusing and one-dimensional self-modulation of waves in nonlinear media. Sov. Phys. JETP 34, 62–69. Zysset, B., P. Beaud, and W. Hodel (1987) Generation of optical solitons in the wavelength region 1.37–1.49 mm. Appl. Phys. Lett. 50, 1027–1029.

4

Cross-Phase Modulation: A New Technique for Controlling the Spectral, Temporal, and Spatial Properties of Ultrashort Pulses P.L. Baldeck, P.P. Ho, and R.R. Alfano

1. Introduction Self-phase modulation (SPM) is the principal mechanism responsible for the generation of picosecond and femtosecond white-light supercontinua. When an intense ultrashort pulse progagates through a medium, it distorts the atomic configuration of the material, which changes the refractive index. The pulse phase is time modulated, which causes the generation of new frequencies. This phase modulation originates from the pulse itself (self-phase modulation). It can also be generated by a copropagating pulse (cross-phase modulation). Several schemes of nonlinear interaction between optical pulses can lead to cross-phase modulation (XPM). For example, XPM is intrinsic to the generation processes of stimulated Raman scattering (SRS) pulses, second harmonic generation (SHG) pulses, and stimulated four-photon mixing (SFPM) pulses. More important, the XPM generated by pump pulses can be used to control, with femtosecond time response, the spectral, temporal, and spatial properties of ultrashort probe pulses. Early studies on XPM characterized induced polarization effects (optical Kerr effect) and induced phase changes, but did not investigate spectral, temporal and spatial effects on the properties of ultrashort pulses. In 1980, Gersten, Alfano, and Belic predicted that Raman spectra of ultrashort pulses would be broadened by XPM (Gersten et al., 1980). The first experimental observation of XPM spectral effects dates to early 1986, when it was reported that intense picosecond pulses could be used to enhance the spectral broadening of weaker pulses copropagating in bulk glasses (Alfano et al., 1986). Since then, several groups have been studying XPM effects generated by ultrashort pump pulses on copropagating Raman pulses (Schadt et al., 1986; Schadt and Jaskorzynska, 1987a; Islam et al., 1987a; Alfano et al., 1987b; Baldeck et al., 1987b–d; Manassah, 1987a, b; Hook et al., 1988), second harmonic pulses (Alfano et al., 1987a; Manassah, 1987c; Manassah and Cockings, 1987; Ho et al., 1988), stimulated four-photon mixing pulses (Baldeck and Alfano, 1987), and probe pulses (Manassah et al., 1985;

118

P.L. Baldeck, P.P. Ho, and R.R. Alfano

Agrawal et al., 1989a; Baldeck et al., 1988a, c). Recently, it has been shown that XPM leads to the generation of modulation instability (Agrawal, 1987; Agrawal et al., 1989b; Schadt and Jaskorzynska, 1987b; Baldeck et al., 1988b, 1988d; Gouveia-Neto et al., 1988a, b), solitary waves (Islam et al., 1987b; Trillo et al., 1988), and pulse compression (Jaskorzynska and Schadt, 1988; Manassah, 1988; Agrawal et al., 1988). Finally, XPM effects on ultrashort pulses have been proposed to tune the frequency of probe pulses (Baldeck et al., 1988a), to eliminate the soliton self-frequency shift effect (Schadt and Jaskorzynska, 1988), and to control the spatial distribution of light in large core optical fibers (Baldeck et al., 1987a). This chapter reviews some of the key theoretical and experimental works that have predicted and described spectral, temporal, and spatial effects attributed to XPM. In Section 2, the basis of the XPM theory is outlined. The nonlinear polarizations, XPM phases, and spectral distributions of coprapagating pulses are computed. The effects of pulse walk-off, input time delay, and group velocity dispersion broadening are particularly discussed. (Additional work on XPM and on SPM theories can be found in Manassah (Chapter 5) and Agrawal (Chapter 3).) Experimental evidence for spectral broadening enhancement, induced-frequency shift, and XPM-induced optical amplification is presented in Section 3. Sections 4, 5, and 6 consider the effects of XPM on Raman pulses, second harmonic pulses, and stimulated four-photon mixing pulses, respectively. Section 7 shows how induced focusing can be initiated by XPM in optical fibers. Section 8 presents measurements of modulation instability induced by cross-phase modulation in the normal dispersion region of optical fibers. Section 9 describes XPMbased devices that could be developed for the optical processing of ultrashort pulses with terahertz repetition rates. Finally, Section 10 summarizes the chapter and highlights future trends.

2. Cross-Phase Modulation Theory 2.1 Coupled Nonlinear Equations of Copropagating Pulses The methods of multiple scales and slowly varying amplitude (SVA) are the two independent approximations used to derive the coupled nonlinear equations of copropagating pulses. The multiple scale method, which has been used for the first theoretical study on induced-phase modulation, is described in Manassah (Chapter 5). The following derivation is based on the SVA approximation. The optical electromagnetic field of two copropagating pulses must ultimately satisfy Maxwell’s vector equation: — ¥ — ¥ E = -m0 and

∂D ∂t

(1a)

4. Cross-Phase Modulation

D = eE + P NL ,

119

(1b)

where e is the medium permitivity at low intensity and PNL is the nonlinear polarization vector. Assuming a pulse duration much longer than the response time of the medium, an isotropic medium, the same linear polarization for the copropagating fields, and no frequency dependence for the nonlinear susceptibility c(3), the nonlinear polarization reduces to P NL (r, z, t) = c (3) E 3 (r, z, t),

(2)

where the transverse component of the total electric field can be approximated by E (r, z, t) =

1 2

{A1 (r, z, t)e i (w t - b z ) + A2 (r, z, t)e i (w t - b z ) + c.c.}. 1

1

2

2

(3)

A1 and A2 refer to the envelopes of copropagating pulses of carrier frequencies w1 and w2, and b1 and b2 are the corresponding propagation constants, respectively. Substituting Eq. (3) into Eq. (2) and keeping only the terms synchronized with w1 and w2, one obtains P NL (r, z, t) = P1NL (r, z, t) + P 2NL (r, z, t),

( c( )( A

P 1NL (r, z, t) = 38 c (3) A1 + 2 A2 P 2NL (r, z, t) =

3 8

3

2

2

2

+ 2 A1

)A e ( )A e (

2

1

2

2

i w1t - b1z )

(4a) ,

(4b)

,

(4c)

i w2t - b2 z )

where P1NL and P2NL are the nonlinear polarizations at frequencies w1 and w2, respectively. The second terms in the right sides of Eqs. (4b) and (4c) are cross-phase modulations terms. Note the factor of 2. Combining Eqs. (1)–(4) and using the slowly varying envelope approximation (at the first order for the nonlinearity), one obtains the coupled nonlinear wave equations: ∂ A1 1 ∂ A1 i ( 2) ∂ 2 A1 w1 2 + + b1 =i n2 A1 + 2 A2 2 ∂ z v g1 ∂t 2 ∂t c

[

2

]A , 1

∂ A2 1 ∂ A2 i ( 2) ∂ 2 A2 w2 2 2 + + b2 =i n2 A2 + 2 A1 A2 , 2 ∂ z v g 2 ∂t 2 ∂t c

[

]

(5a) (5b)

where vgi is the group velocity for the wave i, b (2) is the group velocity i dispersion for the wave i, and n2 = 3c(3)/8n is the nonlinear refractive index. In the most general case, numerical methods are used to solve Eqs. (5). However, they have analytical solutions when the group velocity dispersion temporal broadening can be neglected. Denoting the amplitude and phase of the pulse envelope by a and a, that is,

120

P.L. Baldeck, P.P. Ho, and R.R. Alfano

A1 (t , z) = a1 (t , z)e ia (t ,z ) 1

and A2 (t , z) = a 2 (t , z)e ia

2

( t ,z )

,

(6)

(2) and assuming b (2) 1 ª b 2 ª 0, Eqs. (5a) and (5b) reduce to

∂ a1 = 0, ∂z

(7a)

∂ a1 w1 =i n2 [a 12 + 2a 22 ], ∂z c

(7b)

∂a 2 Ê 1 1 ˆ ∂a 2 +Á = 0, ˜ ∂ z Ë v g 2 v g1 ¯ ∂t ∂a 2 w2 =i n2 [a 22 + 2a 12 ], ∂z c

(7c) (7d)

where t = (t - z/vg1)/T0 and T0 is the 1/e pulse duration. In addition, Gaussian pulses are chosen at z = 0: A1 (t , z = 0) =

P1 -t 2 2 e , Aeff

(8a)

A2 (t , z = 0) =

P2 - (t -td )2 2 e , Aeff

(8b)

where P is the pulse peak power, Aeff is the effective cross-sectional area, and td = Td/T0 is the normalized time delay between pulses at z = 0. With the initial conditions defined by Eqs. (8), Eqs. (5) have analytical solutions when temporal broadenings are neglected: A1 (t , z) =

P1 - t 2 2 ia1 (t ,z ) e e , Aeff

(9a)

A2 (t , z) =

P2 - (t -td - z Lw )2 2 ia2 (t ,z ) e e , Aeff

(9b)

w1 P1 w1 P2 z ˆ˘ 2 È Ê n2 ze - t + p n2 Lw Íerf (t - t d ) - erf t - t d , Ë c Aeff c Aeff Lw ¯ ˙˚ Î (9c) 2 w2 P2 a 2 (t , z) = n2 ze - (t -td - z Lw ) c Aeff w2 P1 z ˆ˘ È Ê (9d) + p n2 Lw Íerf (t ) - erf t , Ë c Aeff Lw ¯ ˙˚ Î a 1 (t , z) =

where Lw = T0/(1/vg1 - 1/vg2) is defined as the walk-off length. Equations (9c) and (9d) show that the phases a1(t, z) of copropagating pulses that overlap in a nonlinear Kerr medium are modified by a cross-phase modulation via the peak power Pjπi. In the case of ultrashort pulses this cross-

4. Cross-Phase Modulation

121

phase modulation gives rise to the generation of new frequencies, as does selfphase modulation. The instantaneous XPM-induced frequency chirps are obtained by differentiating Eqs. (9c) and (9d) according to the instantaneous frequency formula Dw = -∂a/∂t. These are Dw 1 (t , z) = 2

[

]

2 w1 P1 z - t 2 w 1 P2 Lw - (t -td - 2 Lw )2 n2 te 2 n2 e - e - (t -td - z Lw ) , c Aeff T0 c Aeff T0 (10a)

2 w2 P2 z (t - t d - 2 Lw )e - (t -td - z Lw ) n2 c Aeff T0 2 w2 P1 Lw - t 2 +2 n2 e - e - (t - 2 Lw ) , c Aeff T0

Dw 2 (t , z) = 2

[

]

(10b)

where Dw1 = w - w1 and Dw2 = w - w2. The first and second terms on the right sides of Eqs. (10a) and (10b) are contributions arising from SPM and XPM, respectively. It is interesting to notice in Eq. (10) than the maximum frequency chirp arising from XPM is inversely proportional to the group velocity mismatch Lw/T0 = 1/(1/vg1 - 1/vg2) rather than the pump pulse time duration or distance traveled z as for ZPM. Therefore, the time duration of pump pulses does not have to be as short as the time duration of probe pulses for XPM applications. More generally, spectral profiles affected by XPM can be studied by computing the Fourier transform: S (w - w 0 , z) =

1 2p

Ú

+•

-•

a(t , z)e ia (t , z )e i (w -w )t dt , 0

(11)

where |S(w - w0, z)|2 represents the spectral intensity distribution of the pulse. Equation (10) is readily evaluated numerically using fast Fourier transform algorithms. Analytical results of Eqs. (9) take in account XPM, SPM, and group velocity mismatch. These results are used in the Section 2.2 to isolate the specific spectral features arising from the nonlinear interaction of copropagating pulses. Higher-order effects due to group velocity dispersion broadening are discussed in Section 2.3.

2.2 Spectral Broadening Enhancement The spectral evolution of ultrashort pulses interacting in a nonlinear Kerr medium is affected by the combined effects of XPM, SPM, and pulse walk-off. For a negligible group velocity mismatch, XPM causes the pulse spectrum to broaden more than expected from SPM alone. The pulse phase of Eqs. (9c) and (9d) reduces to a i (t , z)

w i (Pi + 2Pj ) - t 2 n2 ze . c Aeff

(12)

122

P.L. Baldeck, P.P. Ho, and R.R. Alfano

The maximum spectral broadening of Gaussian pulses, computed using Eq. (12), is given by Dw i (z) ª

w i (Pi + 2Pj ) z n2 . c Aeff T0

(13)

Thus, the spectral broadening enhancement arising from XPM is given by 2Pj Dw iSPM+XPM = 1+ . Dw iSPM Pi

(14)

Therefore, XPM can be used to control the spectral broadening of probe pulses using strong command pulses. This spectral control is important, for it is based on the electronic response of the interacting medium. It could be turned on and off in a few femtoseconds, which could lead to applications such as the pulse compression of weak probe pulses, frequency-based optical computation schemes, and the frequency multiplexing of ultrashort optical pulses with terahertz repetition rates. The effect of pulse walk-off on XPM-induced spectral broadening can be neglected when wavelengths of pulses are in the low dispersion region of the nonlinear material, the wavelength difference or/and the sample length are small, and the time duration of pulses is not too short. For other physical situations, the group velocity mismatch and initial time delay between pulses affect strongly the spectral shape of interacting pulses (Islam et al., 1987a; Manassah 1987a; Agrawal et al., 1988, 1989a; Baldeck et al., 1988a). Figure 4.1 shows how the spectrum of a weak probe pulse can be affected by the XPM generated by a strong copropagating pulse. The wavelength of the pump pulse was chosen where the pump pulse travels faster than the probe pulse. Initial time delays between pulses at the entrance of the nonlinear medium were selected to display the most characteristic interaction schemes. Figures 4.1a and 4.1b are displayed for reference. They show the probe pulse spectrum without XPM interaction (Figure 4.1a) and after the XPM interaction but for negligible group velocity mismatch (Figure 4.1b). Figure 4.1c is for the case of no initial time delay and total walk-off. The probe spectrum is shifted and broadened by XPM. The anti-Stokes shift is characteristic of the probe and pump pulse walk-off. The probe pulse is blue shifted because it is modulated only by the back of the faster pump pulse. When the time delay is chosen such that the pump pulse enters the nonlinear medium after the probe and has just time to catch up with the probe pulse, one obtains a broadening similar to that in Figure 4.1c but with a reverse Stokes shift (Figure 4.1d). The XPM broadening becomes symmetrical when the input time delay allows the pump pulse not only to catch up with but also to pass partially through the probe pulse (Figure 4.1e). However, if the interaction length is long enough to allow the pump pulse to completely overcome the probe pulse, there is no XPM-induced broadening (Figure 4.1f ). The diversity of spectral features displayed in Figure 4.1 can easily be understood by computing the phase and frequency chirp given by Eqs. (9)

4. Cross-Phase Modulation

123

Figure 4.1. Influence of cross-phase modulation, walk-off, and input time delay on the spectrum of a probe pulse from Eqs. (9) and (11) with P1 << P2. f = 2(w1/c)n2P2Lw, d = z/Lw, and td are the XPM, walk-off, and input time delay parameters, respectively. (a) Reference spectrum with no XPM; i.e., f = 0. (b) XPM in the absence of walk-off; i.e., f = 50 and d = 0. (c) XPM, total walk-off, and no initial time delay; i.e., f = 50, d = -5, and td = 0. (d) XPM and initial time delay to compensate the walk-off; i.e., f = 50, d = -5, and td = 5. (e) XPM and symmetrical partial walk-off; i.e., f = 50, d = -3, and td = 1.5. (f) XPM and symmetrical total walk-off; i.e., f = 50, d = -5, and td = 2.5.

and (10) (Figure 4.2). For reference, Figure 4.2a shows the locations of the pump pulse (solid line) and the probe pulse (dotted line) at the output of the nonlinear sample (case of no initial delay and total walk-off). In this case the XPM phase, which is integrated over the fiber length, has the characteristic shape of an error function whose maximum corresponds to neither the probe pulse maximum nor the pump pulse maximum (Figure 4.2b). The probe pulse (dotted line in Figure 4.2c) sees only the blue part of the frequency chirp (solid line in Figure 4.2c) generated by the pump pulse. As a result, the probe

124

P.L. Baldeck, P.P. Ho, and R.R. Alfano

Figure 4.2. Influence of cross-phase modulation, walk-off, and input time delay on the phase and frequency chirp of a probe pulse. (a) Locations of pump (solid line) and probe (dotted line) at the output of the nonlinear medium for total walk-off and no initial time-delay; i.e., d = -5 and td = 0. (b) XPM phase with a total walk-off and no initial time delay; i.e., f = 50, d = -5, and td = 0. (c) XPM-induced chirp (solid line) with total walk-off and no initial time delay. (Dotted line) Probe pulse intensity. (d) XPM phase with an initial time delay to compensate the walk-off; i.e., f = 50, d = -5, and td = 5. (e) XPM-induced chirp (solid line) with an initial time delay to compensate the walk-off. (Dotted line) Probe pulse intensity. (f) XPM phase and symmetrical partial walk-off; i.e., f = 50, d = -3, and td = 1.5. (g) XPM-induced chirp (solid line) and symmetrical partial walk-off. (Dotted line) Probe pulse intensity. (h) XPM phase and symmetrical total walk-off; i.e., f = 50, d = -5, and td = 2.5. (i) XPM-induced chirp (solid line) and symmetrical total walk-off. (Dotted line) Probe pulse intensity.

4. Cross-Phase Modulation

125

Figure 4.2. (continued)

spectrum is simultaneously broadened and shifted toward the highest frequencies (Figure 4.1c). One should notice that opposite to the SPM frequency chirp, the XPM chirp in Figure 4.2c is not monotonic. The pulse leading edge and trailing edge have a positive chirp and negative chirp, respectively. As a result, dispersive effects (GVD, grating pair, . . .) are different for the pulse front and the pulse back. In the regime of normal dispersion (b(2) > 0), the pulse front would be broadened by GVD while the pulse back would be sharpened. Figures 4.2d and 4.2e show XPM-induced phase and frequency chirp for the mirror image case of Figures 4.2b and 4.2c. The probe spectrum is now shifted toward the smallest frequencies. Its leading edge has a negative frequency chirp, while the trailing edge has a positive one. A positive GVD would compress the pulse front and broaden the pulse back. The case of a partial symmetrical walk-off is displayed in Figures 4.2f and 4.2g. In first approximation, the time dependence of the XPM phase associated with the probe pulse energy is parabolic (Figure 4.2f ), and the frequency chirp is quasi-linear (4.2 g). This is the prime quality needed for the compression of a weak pulse by following the XPM interaction by a grating pair compressor (Manassah, 1988). Figures 4.2h and 4.2i show why there is almost no spectral broadening enhancement when the pump pulse passes completely through the probe pulse (Figure 4.1f ): the part of XPM associated with the probe pulse energy is constant (Figure 4.2h). The probe pulse is phase modulated, but the phase shift is time independent. Therefore, there is neither frequency chirp (Figure 4.2i) nor spectral broadening enhancement by XPM.

126

P.L. Baldeck, P.P. Ho, and R.R. Alfano

Figure 4.3. Influence of self-phase modulation, cross-phase modulation, walk-off, and input time delay on the spectrum of a probe pulse from Eqs. (9) and (11) with P1 = P2. The parameter values in Figure 4.1 are used.

The combined effects of XPM and walk-off on the spectra of weak probe pulses (negligible SPM) have been shown in Figure 4.1 and 4.2. When the group velocity mismatch is large, the spectral broadening is not significant and the above spectral features reduce to a tunable induced-frequency shift of the probe pulse frequency (see Section 3.2). When strong probe pulses are used, the SPM contribution has to be included in the analysis. Figure 4.3 show how the results of Figure 4.1 are modified when the probe power is the same as the pump power, that is, the SPM has to be taken in account. Figure 4.3a shows the spectral broadening arising from the SPM alone. Combined effects of SPM and XPM are displayed in Figures 4.3b to 4.3e with the same initial delays as in Figure 4.1. The SPM contribution to the spectral broad-

4. Cross-Phase Modulation

127

ening is larger than the XPM contribution because the XPM interaction length is limited by the walk-off between pump and probe pulses. The XPM spectral features described in this section have been obtained using first-order approximation of the nonlinear polarization, propagation constant, and nonlinearity in the nonlinear wave equation (Eq. 1). Moreover, plane wave solutions and peak powers below the stimulated Raman scattering threshold have been assumed. For practical purposes it is often necessary to include the effects of (1) first- and second-order group velocity dispersion broadening, b(2) and b(3), (2) induced- and self-steepening, (3) four-wave mixing occurring when pump and probe pulses are coupled through c(3), (4) stimulated Raman scattering generation, (5) the finite time response of the nonlinearity, and (6) the spatial distribution of interacting fields (i.e., induced- and self-focusing, diffraction, Gaussian profile of beams, . . .). In Section 2.3 the combined effect of XPM and group velocity dispersion broadening b(2) is shown to lead to new kinds of optical wave breaking and pulse compression. Some other effects that lead to additional spectral, temporal, and spatial features of XPM are discussed by Agrawal (Chapter 3) and Manassah (Chapter 5).

2.3 Optical Wave Breaking and Pulse Compression due to Cross-Phase Modulation in Optical Fibers When an ultrashort light pulse propagates through an optical fiber, its shape and spectrum change considerably as a result of the combined effect of group velocity dispersion b(2) and self-phase modulation. In the normal dispersion regime of the fiber (l £ 1.3 mm), the pulse can develop rapid oscillations in the wings together with spectral sidelobes as a result of a phenomenon known as optical wave breaking (Tomlinson et al., 1985). In this section it is shown that a similar phenomenon can lead to rapid oscillations near one edge of a weak pulse that copropagates with a strong pulse (Agrawal et al., 1988). To isolate the effects of XPM from those of SPM, a pump-probe configuration is chosen (P2 << P1) so that pulse 1 plays the role of the pump pulse and propagates without being affected by the copropagating probe pulse. The probe pulse, however, interacts with the pump pulse through XPM. To study how XPM affects the probe evolution along the fiber, Eqs. (5a) and (5b) have been solved numerically using a generalization of the beam propagation or the split-step method (Agrawal and Potasek, 1986). The numerical results depend strongly on the relative magnitudes of the length scales Ld and Lw, where Ld = T 20/|b2| is the dispersion length and Lw = vg1vg2T0/|vg1 - vg2| is the walk-off length. If Lw << Ld, the pulses walk off from each other before GVD has an opportunity to influence the pulse evolution. However, if Lw and Ld become comparable, XPM and GVD can act together and modify the pulse shape and spectra with new features. To show these features as simply as possible, a specific case is considered in which Lw/Ld = 0.1 and l1/l2 = 1.2. Both pulses are assumed to propagate

128

P.L. Baldeck, P.P. Ho, and R.R. Alfano

Figure 4.4. Shape and spectrum of probe pulse (left) and pump pulse (right) at z/Ld = 0.4 when the two pulses copropagate in the normal dispersion regime of a single-mode fiber. The parameters are N = 10, Lw/Ld = 0.1, l1/l2 = 1.2, and td = 0. Oscillations near the trailing edge (positive time) of the probe pulse are due to XPMinduced optical wave breaking. (From Agrawal et al., 1988.)

in the normal GVD regime with b1 = b2 > 0. It is assumed that the pump pulse goes faster than the probe pulse (vg1 > vg2). At the fiber input both pulses are taken to be a Gaussian of the same width with an initial delay td between them. First, the case td = 0 is considered, so the two pulses overlap completely at z = 0. Figure 4.4 shows the shapes and spectra of the pump and probe pulses at z/Ld = 0.4 obtained by solving Eqs. (5a) and (5b) numerically with N = (g 1P1Ld)0.5 = 10. For comparison, Figure 4.5 shows the probe and pump spectra under identical conditions but without GVD effects (b1 = b2 = 0). The pulse shapes are not shown since they remain unchanged when the GVD effects are excluded.

4. Cross-Phase Modulation

129

Figure 4.5. Spectra of probe and pump pulses under conditions identical to those of Figure 4.4 but without the GVD effects (b1 = b2 = 0). Pulse shapes are not shown as they remain unchanged. (From Agrawal et al., 1988.)

From a comparison of Figures 4.4 and 4.5, it is evident that GVD can substantially affect the evolution of features expected from SPM or XPM alone. Consider first the pump pulse for which XPM effects are absent. The expected from dispersive SPM for N = 10. With further propagation, the pump pulse eventually develops rapid oscillations in the wings as a result of conventional SPM-induced optical wave breaking. Consider now the probe pulse for which SPM effects are absent and probe pulse evolution is governed by dispersive XPM. In absence of GVD, the pulse shape would be a narrow Gaussian centered at t = 4 (the relative delay at the fiber output because of group velocity mismatch). The GVD effects not only broaden the pulse considerably but also induce rapid oscillations near the trailing edge of the probe pulse. These oscillations are due to XPM-induced optical wave breaking. To understand the origin of XPM-induced optical wave breaking, it is useful to consider the frequency chirp imposed on the probe pulse by the copropagating pulse. As there is total walk-off and no initial delay, maximum chirp occurs at the center of the probe pulse. Since the chirp is positive, blueshifted components are generated by XPM near the pulse center. As a result of the normal GVD, the peak of the probe pulse moves slower than its tails. Since the peak lags behind as the probe pulse propagates, it interferes with the trailing edge. Oscillations seen near the trailing edge of the probe pulse in Figure 4.4 result from such an interference. Since the basic mechanism is analogous to the optical wave-breaking phenomenon occurring in the case of dispersive XPM, we call it XPM-induced optical wave breaking. In spite of the identical nature of the underlying physical mechanism, optical wave breaking exhibits different qualitative features in the XPM case

130

P.L. Baldeck, P.P. Ho, and R.R. Alfano

compared with the SPM case. The most striking difference is that the pulse shape is asymmetric with only one edge developing oscillations. For the case shown in Figure 4.4 oscillations occur near the trailing edge. If the probe and pump wavelengths were reversed so that the pump pulse moved slower than the probe pulse, oscillations would occur near the leading edge since the pump pulse would interact mainly with that edge. In fact, in that case the shape and the spectrum of the probe pulse are just the mirror images of those shown in Figures 4.4 and 4.5. The effect of initial delay between probe and pump pulses is now investigated. The effect of initial delay on XPM-induced spectral broadening has been discussed in the dispersionless limit (b1 = b2 = 0) in Section 2.2. For example, if the pump pulse is delayed by the right amount so that it catches

Figure 4.6. Probe shape and spectrum with (top) and without (bottom) the GVD effects under conditions identical to those of Figure 4.5 except that td = -2. Note the important effect on pulse evolution of the initial time delay between the pump and probe pulses. (From Agrawal et al., 1988.)

4. Cross-Phase Modulation

131

up with the probe pulse at the fiber output, the probe spectrum is just the mirror image of that shown in Figure 4.4, exhibiting a red shift rather than a blue shift. Futhermore, if td is adjusted such that the pump pulse catches up with the probe pulse halfway through the fiber, the probe spectrum is symmetrically broadened since the pump walks through the probe in a symmetric manner. Our numerical results show that the inclusion of GVD completely alters this behavior. Figure 4.6 shows the probe shape and spectrum under conditions identical to those of Figure 4.4 except that the probe pulse is advanced (td = -2) such that the pump pulse would catch it halfway through the fiber in the absence of GVD effects. The lower row shows the expected behavior in the dispersionless limit, showing the symmetrical spectral broadening in this case of symmetrical walk-off. A direct comparison reveals how much the presence of GVD can affect the SPM effects on the pulse evolution. In particular, both the pulse shape and spectra are asymmetric. More interestingly, the probe pulse is compressed, in sharp contrast to the case of Figure 4.4, where GVD led to a huge broadening. This can be understood qualitatively from Eq. (10). For the case shown in Figure 4.6, the XPMinduced chirp is negative and nearly linear across the trailing part of the probe pulse. Because of this chirp, the traveling part is compressed as the probe pulse propagates inside the fiber. Experimental observation of XPM-induced optical wave breaking would require the use of femtosecond pulses. This can be seen by noting that for picosecond pulses with T0 = 5–10 ps, typically Ld ª 1 km while Lw ª 1 m even if the pump-probe wavelengths differ by as little as 10 nm. By contrast, if T0 = 100 fs, both Ld and Lw become comparable (ª10 cm), and the temporal changes in the probe shape discussed here can occur in a fiber less than a meter long. Pulses much shorter than 100 fs should also not be used since higher-order nonlinear effects such as self-steepening and a delayed nonlinear response then become increasingly important. Although these effects are not expected to eliminate the phenomenon of XPM-induced optical wave breaking, they may interfere with the interpretation of experimental data.

3. Pump-Probe Cross-Phase Modulation Experiments Cross-phase modulation is intrinsic to numerous schemes of ultrashort pulse interaction. The first observation of spectral effects arising from XPM was reported using a pump-probe scheme (Alfano et al., 1986). The phase modulation generated by the infrared pulse at the probe wavelength was referred to as an induced-phase modulation (PM). More recently, the inducedfrequency shift and spectral broadening enhancement of picosecond probe pulses have been observed using optical fibers as nonlinear media (Baldeck et al., 1988a; Islam et al., 1987a, b). Pump-probe experiments on XPM are of prime importance for they could lead to applications for pulse compres-

132

P.L. Baldeck, P.P. Ho, and R.R. Alfano

sion, optical communication, and optical computation purposes. Results of the pump-probe experiments on XPM are discussed in this section.

3.1 Spectral Broadening Enhancement by Cross-Phase Modulation in BK-7 Glass The possibility of enhancing the spectral broadening of a probe pulse using a copropagating pump pulse was first observed experimentally in early 1986 (Alfano et al., 1986). The spectral broadening of a weak 80-mJ picosecond 530-nm laser in BK-7 glass was enhanced over the entire spectral band by the presence of an intense millijoule picosecond 1060-nm laser pulse. The spectral distributions of the self-phase modulation and the cross-phase modulation signals were found to be similar. The dominant enhancement mechanism for the induced supercontinuum was determined to be a cross-phase modulation process, not stimulated four-photon scattering. The experimental setup is shown in Figure 4.7. A single 8-ps laser pulse at 1060 nm generated from a mode-locked glass laser system was used as the pump beam. Its second harmonic was used as the probe beam. These pulses at the primary 1060-nm and the second harmonic 530-nm wavelengths were weakly focused into a 9-cm-long BK-7 glass. A weak supercontinuum signal was observed when both 530- and 1060-nm laser pulses were sent through the sample at the same time. This signal could arise from the IPM process and/or stimulated four-photon parametric generation (FPPG). In this induced supercontinuum experiment, the 530-nm laser pulse intensity was kept nearly constant with a pulse energy of about 80 mJ. The primary 1060-nm laser pulse energy was a controlled variable changing from 0 to 2 mJ. Filters were used to adjust the 1060-nm pump-laser pump intensity. The output beam was separated into three paths for diagnosis. The output beam along path 1 was imaged onto the slit of a 0.5-m JarrelAsh spectrograph to separate the contributions from the possible different mechanisms for the supercontinuum by analyzing the spatial distribution of the spectrum from phase modulation and stimulated four-photon scattering processes. In this spectrograph measurement, films were used to measure the spatial distribution of the supercontinuum spectrum and a photomultiplier tube was used to obtain quantitative reading. To distinguish different contributions from either phase modulation or stimulated four-photon scattering, geometric blocks were arranged in the path for the selection of a particular process. An aperture of 6 mm diameter was placed in front of the entrance slit of the spectrograph to measure the signal contributed phase modulation, while an aluminum plate of 7 mm width was placed in front of the spectrograph entrance slit to measure the l = 570 nm contribution. The beam along path 2 was directed into a spectrometer with an optical multichannel analyzer to measure the supercontinuum spectral intensity distribution. The spectrum was digitized, displayed, and stored in 500 channels

4. Cross-Phase Modulation

133

Figure 4.7. Schematic diagram of the experimental arrangment for measuring the spectral broadening enhancement of probe pulses by induced-phase modulation. F1: Hoya HA30 (0.03%), R72 (82%), Corning 1-75 (1%), 1-59 (15%), 0-51 (69%), 3-75 (80%). The numbers in parentheses correspond to the transmittivity at 1054 nm. All these color filters have about 82% transmittivity at 527 nm. F2: 1-75 + 3-67 for Stokes side measurements; F2: 1-75 + 2 (5-57) for anti-Stokes side measurements; F3: neutral density filters; F: ND3 + 1-75; D1, D2; detectors; M: dielectric-coated mirror; BS: beam splitter. (From Alfano et al., 1986.)

as a function of wavelength. The beam along path 3 was delayed and directed into a Hamamatsu Model C1587 streak camera to measure the temporal distribution of the laser pulse and induced supercontinuum. The duration of the induced supercontinuum with a selected 10-nm bandwidth was measured to be about the same as the incident laser pulse duration. Experimental results for the spectral distribution of induced supercontinuum and supercontinuum are displayed in Figure 4.8. More than 20 laser shots for each data point in each instance have been normalized and smoothed. The average gain of the induced supercontinuum in a BK-7 glass from 410- to 660-nm wavelength was about 11 times that of the supercontinuum. In this instance, both the 530- and 1060-nm laser pulse energies were maintained nearly constant: 80 mJ for 530 nm and 2 mJ for 1060 nm. In this experiment, the 530-nm laser pulse generated a weak supercontinuum and the intense 1060-nm laser pulse served as a catalyst to enhance the super-

134

P.L. Baldeck, P.P. Ho, and R.R. Alfano

Figure 4.8. Intensities of the induced ultrafast supercontinuum pulse (IUSP) and the ultrafast supercontinuum pulse (USP). Each data point was an average of about 20 laser shots and was corrected for the detector, filter, and spectrometer spectral sensitivity. (D) IUSP (F1: 3-75); (䊊) USP from 527 nm (F1: HA30). USP from 1054 nm, which is not shown here was ª1% of the IUSP signal. The measured 527-nm probe pulse was about 5 ¥ 10 counts on this arbitrary unit scale. The error bar of each data point is about ±20%. (From Alfano et al., 1986.)

continuum in the 530-nm pulse. The supercontinuum generated by the 1060nm pulse alone in this spectral region was less than 1% of the total induced supercontinuum. The spectral shapes of the induced supercontinuum pulse and the supercontinuum pulse in Figure 4.8 are similar. Use of several liquid samples such as water, nitrobenzene, CS2, and CCl4 has also been attempted to obtain the induced supercontinuum. There was no significant (twofold) enhancement from all other samples that we tested. A plot of the intensity dependence of the induced supercontinuum is displayed in Figure 4.9 as a function of the 1060-nm pump pulse energy. The wavelengths plotted in Figure 4.9 were l = 570 nm for the Stokes side and l = 498 nm for the anti-Stokes side. The 530-nm pulse energy was set at 80 ± 15 mJ. The induced supercontinuum increased linearly as the added 1060-nm laser pulse energy was increased from 0 to 200 mJ. When the 1060-nm pump pulse was over 1 mJ, the supercontinuum enhancement reached a plateau and saturated at a gain factor of about 11 times over the supercontinuum inten-

4. Cross-Phase Modulation

135

Figure 4.9. Dependence of the IUSP signal on the intensity of the 1.06-mm pump pulse. (䊊) Stokes side at l = 570 nm; (D) anti-Stokes side at l = 498 nm. The error bars of the anti-Stokes side were similar to those of the Stokes side. The solid line is a guide for the eye. The vertical axis is the normalized IIUSP/I527 nm. (From Alfano et al., 1986.)

sity generated by only the 530-nm pulse. This gain saturation may be due to the trailing edge of the pulse shape function being maximally distorted when the primary pulse intensity reaches a certain critical value. This implies a saturation of the PM spectral distribution intensity when the pumped primary pulse energy is above 1 mJ, as shown in Figure 4.9. Since the supercontinuum generation can be due to the phase modulation and/or the stimulated four-photon scattering processes, it is important to distinguish between these two different contributions to the induced supercontinuum signal. Spatial filtering of the signal was used to separate the two main contributions. The induced supercontinuum spectrum shows a spatial spectral distribution similar to that of the conventional supercontinuum. The collinear profile that is due to the phase modulation has nearly the same spatial distribution as the incident laser pulse. Two emission wings at noncollinear angles correspond to the stimulated four-photon scattering continuum arising from the phase-matching condition of the generated wavelengths emitted at different angles from the incident laser beam direction. Using a

136

P.L. Baldeck, P.P. Ho, and R.R. Alfano

Figure 4.10. Dependence of Is (PM) and Is (FPPG) at l = 570 nm or the intensity of the 1054-nm pump laser pulse. (䊊) PM; (D) FPPG. The measured signal has been normalized with the incident 527-nm pulse energy. The error bar of each data point is about ±20% of the average value. (From Alfano et al., 1986.)

photomultiplier system and spatial filtering, quantitative measurements of the induced supercontinuum contributions from the collinear PM and the noncollinear stimulated four-photon scattering parts were obtained (Figure 4.10). These signals, measured at l = 570 nm from the collinear PM and the noncollinear parts of the induced supercontinuum, are plotted as a function of the pump pulse energy. There was little gain from the contribution of the stimulated four-photon scattering process over the entire pulse-energydependent measurement as shown in Figure 4.10. The main enhancement of the induced supercontinuum generation is consequently attributed to the PM mechanism, which corresponds to the collinear geometry. Another possible mechanism for the observed induced supercontinuum could be associated with the enhanced self-focusing of the second harmonic pulse induced by the primary pulse. There was no significant difference in the spatial intensity distribution of the 530-nm probe beam with and without the added intense 1060-nm pulse. In this experiment the spectral broadening of 530-nm pulses was enhanced by nonlinear interaction with copropagating strong infrared pulses in a BK-

4. Cross-Phase Modulation

137

7 glass sample. The spectral change has been found to arise from a phase modulation process rather than a stimulated four-photon mixing process. It is in good agreement with predictions of the induced-phase modulation theory. This experiment showed the first clear evidence of a cross-phase modulation spectral effect.

3.2 Induced-Frequency Shift of Copropagating Pulses Optical fibers are convenient for the study of nonlinear optical processes. The optical energy is concentrated into small cross section (typically 10-7 cm2) for long interaction lengths. Thus, large nonlinear effects are possible with moderate peak powers (10–104 W). Optical fibers appear to be an ideal medium in which to investigate XPM effects. The first pump probe experiment using picosecond pulses propagating in optical fibers demonstrated the importance of the pulse walk-off in XPM spectral effects (Baldeck et al., 1988a). It was shown that ultrashort pulses that overlap in a nonlinear and highly dispersive medium undergo a substantial shift of their carrier frequencies. This new coherent effect, which was referred to as an induced-frequency shift, resulted from the combined effect of cross-phase modulation and pulse walk-off. In the experiment, the induced-frequency shift was observed by using strong infrared pulses that shifted the frequency of weak picosecond green pulses copropagating in a 1-m-long single-mode optical fiber. Tunable red and blue shifts were obtained at the fiber output by changing the time delay between infrared and green pulses at the fiber input. A schematic of the experimental setup is shown in Figure 4.11. A modelocked Nd: YAG laser with a second harmonic crystal was used to produce

Figure 4.11. Experimental setup used to measure the induced-frequency shift of 532-nm pulses as a function of the time delay between pump and probe pulses at the optical fiber input. Mirrors M1 and M2 are wavelength selective; i.e., they reflect 532-nm pulses and transmit 1064-nm pulses. (From Baldeck et al., 1988a.)

138

P.L. Baldeck, P.P. Ho, and R.R. Alfano

33-ps infrared pulses and 25-ps green pulses. These pulses were separated using a Mach-Zehnder interferometer delay scheme with wavelengthselective mirrors. The infrared and green pulses propagated in different interferometer arms. The optical path of each pulse was controlled using variable optical delays. The energy of infrared pulses was adjusted with neutral density filters in the range 1 to 100 nJ while the energy of green pulses was set to about 1 nJ. The nonlinear dispersive medium was a 1-m-long singlemode optical fiber (Corguide of Corning Glass). This length was chosen to allow for total walk-off without losing control of the pulse delay at the fiber output. The group velocity mismatch between 532 and 1064-nm pulses was calculated to be about 76 ps/m in fused silica. The spectrum of green pulses was measured using a grating spectrometer (1 meter, 1200 lines/mm) and an optical multichannel analyzer (OMA2). The spectra of green pulses propagating with and without infrared pulses are plotted in Figure 4.12. The dashed spectrum corresponds to the case of green pulses propagating alone. The blue-shifted and red-shifted spectra are those of green pulses copropagating with infrared pulses after the input delays were set at 0 and 80 ps, respectively. The main effect of the nonlinear interaction was to shift the carrier frequency of green pulses. The inducedwavelength shift versus the input delay between infrared and green pulses is

Figure 4.12. Cross-phase modulation effects on spectra of green 532-nm pulses. (a) Reference spectrum (no copropagating infrared pulse). (b) Infrared and green pulses overlapped at the fiber input. (c) Infrared pulse delayed by 80 ps at the fiber input. (From Baldeck et al., 1988a.)

4. Cross-Phase Modulation

139

Figure 4.13. Induced wavelength shift of green 532-nm pulses as a function of the input time delay between 532-nm pulses and infrared 1064-nm pulses at the input of a 1-m-long optical fiber. ( ) Experimental points. The solid line is the theoretical prediction from Eq. (3.3). (From Baldeck et al., 1988a.)

plotted in Figure 4.13. The maximum induced-wavelength shift increased linearly with the infrared pulse peak power (Figure 4.14). Hence, the carrier wavelength of green pulses could be tuned up to 4 Å toward both the red and blue sides by varying the time delay between infrared and green pulses at the fiber input. The solid curves in Figures 4.13 and 4.14 are from theory. When weak probe pulses are used the SPM contribution can be neglected in Eqs. (9) and (10). Thus, nonlinear phase shifts and frequency chirps are given by a 1 (t , z) ª p dw 1 (t , z) ª - 2

w1 P2 z ˆ˘ È Ê n2 Lw Íerf (t - t d ) - erf t - t d + , Ë ¯˙ c Aeff L Î w ˚

[

]

2 w1 P2 Lw - (t -td )2 n2 e - e - (t -td - z Lw ) . c Aeff T0

(15) (16)

When the pulses coincide at the fiber entrance (td = 0) the point of maximum phase is generated ahead of the green pulse peak because of the group velocity mismatch (Eq. 15). The green pulse sees only the trailing part of the XPM profile because it travels slower than the pump pulse. This leads to a blue induced-frequency shift (Eq. 16). Similarly, when the initial delay is set at 80 ps, the infrared pulse has just sufficient time to catch up with the

140

P.L. Baldeck, P.P. Ho, and R.R. Alfano

Figure 4.14. Maximum induced wavelength shift of 532-nm pulses versus the peak power of infrared pump pulses. ( ) Experimental points. The solid line is the theoretical prediction from Eq. (3.4). (From Baldeck et al., 1988a.)

green pulse. The green pulse sees only the leading part of the XPM phase shift, which gives rise to a red induced-frequency shift. When the initial delay is about 40 ps, the infrared pulse has time to pass entirely through the green pulse. The pulse envelope sees a constant dephasing and there is no shift of the green spectrum (Figure 4.13). Equations (15) and (16) can be used to fit our experimental data shown in Figures 4.13 and 4.14. Assuming that the central part of the pump pulses provides the dominant contribution to XPM, we set t = 0 in Eq. (16) and obtain dw 1 (t , z) ª

[

]

2 w1 P2 Lw - (t -td )2 n2 e - e - (t -td + z Lw ) . c Aeff T0

(17)

The maximum induced-frequency shift occurs at td = d = z/Lw and is given by Dw max =

w1 P2 Lw n2 . c Aeff T0

(18)

Equations (17) and (18) are plotted in Figures 4.13 and 4.14, respectively. There is very good agreement between this simple analytical model and experimental data. It should be noted that only a simple parameter (i.e., the

4. Cross-Phase Modulation

141

infrared peak power at the maximum induced-frequency shift) has been adjusted to fit the data. Experimental parameters were l = 532 nm, T0 = 19.8 ps (33 ps FWHM), Lw = 26 cm, and d = 4. We have shown experimentally and theoretically that ultrashort optical pulses that overlap in a nonlinear and highly dispersive medium can undergo a substantial shift of their carrier frequency. This induced-frequency shift has been demonstrated using strong infrared pulses to shift the frequency of copropagating green pulses. The results are well explained by an analytical model that includes the effect of cross-phase modulation and pulse walk-off. This experiment led to a conclusive observation of XPM spectral effects.

3.3 XPM-Induced Spectral Broadening and Optical Amplification in Optical Fibers This section presents additional features that can arise from the XPM interaction between a pump pulse at 630 nm and a probe pulse at 532 nm. With this choice of wavelengths, the group velocity dispersion between the pump pulse and the probe pulse is reduced and the XPM interaction enhanced. The spectral width and the energy of the probe pulse were found to increase in the presence of the copropagating pump pulse (Baldeck et al., 1988c). A schematic of the experimental setup is shown in Figure 4.15. A modelocked Nd: YAG laser with a second harmonic crystal was used to produce pulses of 25-ps duration at 532 nm. Pump pulses were obtained through stimulated Raman scattering by focusing 90% of the 532-nm pulse energy into a 1-cm cell filled with ethanol and using a narrowband filter centered at 630 nm. Resulting pump pulses at 630 nm were recombined with probe pulses and coupled into a 3-m-long single-mode optical fiber. Spectra of probe pulses were recorded for increasing pump intensities and varying input time delays between pump and probe pulses.

Figure 4.15. Experimental setup for generating copropagating picosecond pulses at 630 and 532 nm. (From Baldeck et al., 1988c.)

142

P.L. Baldeck, P.P. Ho, and R.R. Alfano

Figure 4.16. Cross-phase modulation effects on the spectrum of a probe picosecond pulse. (Dashed line) Reference spectrum without XPM. (Solid line) With XPM and no time delay between pump and probe pulses at the optical fiber input. (From Baldeck et al., 1988c.)

With negative delays (late pump at the optical fiber input), the spectrum of the probe pulse was red shifted as in the 1064 nm/532 nm experiment (Figure 4.12). A new XPM effect was obtained when both pulses entered the fiber simultaneously. The spectrum of the probe pulse not only shifted toward blue frequencies as expected but also broadened (Figure 4.16). An spectral broadening as wide as 10 nm could be induced, which was, surprisingly, at least one order of magnitude larger than predicted by the XPM theory. As shown in Figure 4.16, the probe spectrum extended toward the blue-shifted frequencies with periodic resonant lines. These lines could be related to modulation instability sidelobes that have been predicted theoretically to occur with cross-phase modulation (see Section 8). The optical amplification of the probe pulse is another new and unexpected feature arising from the XPM interaction. Pump power-dependent gain factors of 3 or 7 were measured using probe pulses at 532 nm and pump pulses at 630 or 1064 nm, respectively. Figure 4.17 shows the dependence of the XPMinduced gain for the probe pulse at 532 nm with the input time delay between the probe pulse and the pump pulse at 630 nm. The shape of the gain curve corresponds to the overlap function of pump and probe pulses. Figure 4.1